Discontinuous Galerkin Methods with Optimal $L^2$ Accuracy for One Dimensional Linear PDEs with High Order Spatial Derivatives

Abstract

In this paper, we formulate and analyze discontinuous Galerkin (DG) methods to solve several partial differential equations (PDEs) with high order spatial derivatives, including the heat equation, a third order wave equation, a fourth order equation and the linear Schrödinger equation in one dimension. Following the idea of local DG methods, we first rewrite each PDE into its first order form and then apply a general DG formulation. The numerical fluxes are introduced as linear combinations of average values of fluxes, and jumps of the solution as well as the auxiliary variables at cell interfaces. The main focus of the present work is to identify a sub-family of the numerical fluxes by choosing the coefficients in the linear combinations, so the solution and some auxiliary variables of the proposed DG methods are optimally accurate in the $L^2$ norm. In our analysis, one key component is to design some special projection operator(s), tailored for each choice of numerical fluxes in the sub-family, to eliminate those terms at cell interfaces that would otherwise contribute to the sub-optimality of the error estimates. Our theoretical findings are validated by a set of numerical examples.

1 Introduction

In this paper, we propose discontinuous Galerkin (DG) methods with optimal accuracy for solving several partial differential equations (PDEs) with high order spatial derivatives in one dimension. They include:

• The even order equations

  • the heat equation

    $$\begin{aligned} u_t-u_{xx}=0; \end{aligned}$$

    (1.1)

  • the fourth order equation

    $$\begin{aligned} u_t+u_{xxxx}=0; \end{aligned}$$

    (1.2)

  • the equation of an arbitrary even order,

    $$\begin{aligned} u_t+(-1)^{\frac{n}{2}}u^{(n)}_x=0, \end{aligned}$$

    (1.3)

    with n being any positive even integer. Here $u^{(n)}_x$ denotes the n-th derivative of u with respect to x.

• The third order wave equation

  $$\begin{aligned} u_t+u_{xxx}=0; \end{aligned}$$

  (1.4)

• The linear Schrödinger equation

  $$\begin{aligned} iu_t+u_{xx}=0. \end{aligned}$$

  (1.5)

The boundary conditions are assumed to be periodic. These equations provide classical mathematical models for many important physical and engineering applications. The heat equation (1.1) models the heat conduction. The third order wave equation belongs to the KdV-type equations, which describe the propagation of waves in a variety of dispersive media [3]. The fourth order problem (1.2) has wide applications in modeling of thin beams and plates, strain gradient elasticity, and phase separation in binary mixtures [20]. The last equation we consider is the linear Schrödinger equation which has broad applications in fluid dynamics, nonlinear optics, and plasma physics [4, 18].

DG methods are a class of finite element methods using a completely discontinuous piecewise polynomial space for the numerical solution and test functions. The first DG method was introduced by Reed and Hill [26] for the linear neutron transport equation. It was then developed for time-dependent nonlinear hyperbolic conservation laws, coupled with high order Runge-Kutta time discretizations, by Cockburn et al. in [12,13,14,15, 17]. DG methods have grown their popularity over the past few decades in many applications due to their flexibility with meshing and local approximations, their compactness and high parallel efficiency, their excellent dispersion property in wave simulations, and their suitability for various types of differential equations (see, e.g. [24, 27]). Particularly the methods find their success in solving time-dependent PDEs with high order spatial derivatives, with several ideas proposed, such as the penalty methods [2, 21] that add penalty terms at cell interfaces for numerical stability; the local DG (LDG) methods [1, 16, 20, 31] that are formulated based on the first order form of the equations by introducing auxiliary variables; the hybrid DG (HDG) methods [7, 8, 11, 19] that, in addition to working with the first order form as in LDG methods, also include the trace of some variables on mesh skeletons as the additional unknowns in order to create opportunity for the ultimate implementation efficiency; the ultra-weak DG (UWDG) methods [10] that are based on repeated applications of integration by parts with all spatial derivatives moved to the test function in the weak formulation; the direct DG (DDG) methods [25] that are based on a more standard weak formulation of a second order diffusive operator; the conservative DG methods [5] that are based on certain weak formulation derived from repeated integration by parts for the dispersive term and a globally defined projection to preserve the energy for the KdV equation. All the methods mentioned above except the penalty type depend on the suitable design of one (such as in DDG methods) or multiple numerical fluxes (e.g. in LDG, HDG, and UWDG methods) in order to achieve numerical stability, (sub-)optimal accuracy of one (e.g. in DDG and UWDG methods) or more (e.g. in LDG and HDG methods) unknowns, and even implementation efficiency (e.g. in LDG and HDG methods).

In this work, we design DG methods for solving the PDEs (1.1)–(1.5) with high order spatial derivatives. Just as in LDG methods, we start with the first order form of each PDE, and apply a general DG formulation. The numerical fluxes are introduced as some linear combinations of average values of fluxes and jumps of the solution as well as the auxiliary variables at cell interfaces, and they involve a set of parameters as the expansion coefficients. Standard LDG methods can be obtained if one takes special values of these parameters to ensure that all the auxiliary variables can be expressed locally in terms of the original unknown. Instead of requiring such local elimination property, we here identify a sub-family of these parameters so that the respective DG methods are optimal in accuracy for the original unknown and also for some auxiliary variables. The LDG methods in [6, 16, 20, 29,30,31,32] for solving Eqs. (1.1)–(1.5) are special cases of what proposed here. Similar numerical fluxes as well as a special sub-family (termed $\alpha \beta$-fluxes) are investigated in [9] to solve the one-dimensional two-way wave problem, and they lead to a class of $L^2$ stable and optimally accurate DG methods.

The optimal error estimates of our proposed methods reply on two ingredients. One is the energy equations related to numerical stability and the other is the projection operators that measure the approximation property of the discrete spaces and ensure the optimality of the accuracy. For PDEs with high order spatial derivatives, it was discovered in [20, 30] that, to prove optimal accuracy, more than one energy equation may be needed in the presence of the auxiliary variables within the first order forms. These energy equations in general are not trivial to find. Fortunately the stability analyses for LDG methods have partially addressed this aspect for the even order PDEs in [20] (see Remark 3.6) and for the third order wave equation in [30] , and for the linear Schrödinger equation in [30] (see Remark 5.2). As for the projection operators, we will follow [9] and work with some similar type of projection operators that are tailored for each choice of numerical fluxes in the identified special sub-family, to eliminate those terms at cell interfaces that would otherwise contribute to the sub-optimality of the error estimates.

For the proposed semi-discrete DG methods, we further apply in time the implicit version of the spectral deferred correction (SDC) methods [22]. Such methods can be easily constructed to have arbitrary order of accuracy, and they only need to store the numerical solution at the n-th time level in order to compute the solution at $(n+1)$-th time level. In [28], Xia et al demonstrated that the implicit SDC methods provide efficient time discretizations for the LDG methods to solve PDEs with high order spatial derivatives.

The remaining of this paper is organized as follows. In Sect. 2, notations are introduced for meshes, discrete spaces and projection operators. In Sects. 3–5, we propose and analyze DG methods for even-order equations (1.1)–(1.3), the third order wave equation (1.4), and the linear Schrödinger equation (1.5), respectively. The presentation in each section starts with the method, energy relations for numerical stability, and error estimates. Parameters in the numerical fluxes are identified for the $L^2$ stability and for the optimality of the accuracy of the proposed methods. In Sect. 6, numerical examples are presented to verify our theoretical results. The concluding remarks are given in Sect. 7.

2 Discrete Spaces and Projections

Let the computational domain be $\Omega =[x_{min},x_{max}]$, with a partition or mesh $x_{min}=x_{\frac{1}{2}}<x_{\frac{3}{2}}<\cdots <x_{N+\frac{1}{2}}=x_{max}$. Let $I_{j}=[x_{j-\frac{1}{2}},x_{j+\frac{1}{2}}]$ denote an element with the length $\Delta x_j=x_{j+\frac{1}{2}}-x_{j-\frac{1}{2}}$, and $h=\max _{1\le j\le N} \Delta x_j$. Define ${\mathcal {T}}_h=\{ I_j: j=1, 2, \ldots , N\}$. The following discrete space will be used,

$$\begin{aligned} {\mathcal {V}}_h^k=\{{v} : {v}|_{I_j} \in P^k(I_j), \forall I_j \in {\mathcal {T}}_h\}, \end{aligned}$$

(2.1)

where $P^{k}(I_j)$ is the space of polynomials with degree at most k in $I_j$. For any $v\in {\mathcal {V}}_h^k$, $v^+_{j+\frac{1}{2}}$ and $v^-_{j+\frac{1}{2}}$ denote the limit values of v at $x_{j+\frac{1}{2}}$ from the right element $I_{j+1}$ and from the left element $I_j$, respectively. As usual, $\{v\}_{j+\frac{1}{2}}=\frac{1}{2}(v^+_{j+\frac{1}{2}}+v^-_{j+\frac{1}{2}})$ and $[v]_{j+\frac{1}{2}}=(v^+_{j+\frac{1}{2}}-v^-_{j+\frac{1}{2}})$ represent, respectively, the average and the jump of the function v at $x_{j+\frac{1}{2}}$ for any j. We also define

$$\begin{aligned} F_1(\varphi ,\psi ,\beta _1)&=\{\varphi \}+\alpha [\varphi ]+\beta _1[\psi ],\end{aligned}$$

(2.2)

$$\begin{aligned} F_2(\psi ,\varphi ,\beta _2)&=\{\psi \}-\alpha [\psi ]+\beta _2[\varphi ], \end{aligned}$$

(2.3)

at cell interfaces, where $\alpha$, $\beta _1$, $\beta _2$ are constants that are O(1) and they will be specified when being used, while $\psi ,\varphi \in H^1({\mathcal {T}}_h)$ are piecewise-defined with respect to the mesh and have well-defined left and right traces at mesh nodes. Note that both $F_1$ and $F_2$ involve the same parameter $\alpha$, and we omit the $\alpha$-dependence in notation for brevity.

For square integrable functions on a given domain K, the standard notations are used for the inner product and the $L^2$ norm, namely,

$$\begin{aligned} (v,w)_{K}:=\int _{K} vwdx, \quad \Vert v\Vert _K:=\sqrt{(v,v)_{K}}, \quad \forall v, w\in L^2(K). \end{aligned}$$

(2.4)

When $K=\Omega$, we also write (v, w) and $\Vert v\Vert$.

Next, we will introduce the standard $L^2$ projection $P_h$, that projects a function $v\in L^2(\Omega )$ onto the discrete space ${\mathcal {V}}_h^k$, and the Gauss-Radau projections $P_h^\pm$, that project a function $v\in H^{1}({\mathcal {T}}_h)$ onto ${\mathcal {V}}_h^k$. They are defined as follows,

$$\begin{aligned} \int _{I_j}(P_hv-v)wdx&=0, \quad \forall w\in P^k(I_j), \end{aligned}$$

(2.5)

$$\begin{aligned} \int _{I_j}(P_h^+v-v)wdx&=0, \quad \forall w\in P^{k-1}(I_j) \ \text {and} \ P_h^+ v\left( x_{j-\frac{1}{2}}^+\right) ={v\left( x_{j-\frac{1}{2}}^+\right) }, \end{aligned}$$

(2.6)

$$\begin{aligned} \int _{I_j}(P_h^-v-v)wdx&=0, \quad \forall w\in P^{k-1}(I_j) \ \text {and} \ P_h^- v\left( x_{j+\frac{1}{2}}^-\right) ={v\left( x_{j+\frac{1}{2}}^-\right) }, \end{aligned}$$

(2.7)

for any $j=1, \ldots , N$, and have the following approximation property:

$$\begin{aligned} \Vert v-\pi _hv\Vert ^2+h\sum \limits _{j}((v-\pi _hv)_{j+\frac{1}{2}}^{\pm }) ^2\leqslant C_*h^{2k+2}\Vert v\Vert ^2_{{H^{k+1}(\Omega )}},\quad \forall v\in H^{k+1}(\Omega ), \end{aligned}$$

(2.8)

where $\pi _h=P_h^\pm$ or $P_h$, and $C_*$ is a positive constant depending on k but not on h or v. Throughout, the standard notations are used for the Sobolev space $H^{k+1}(\Omega )$ and its norm $\Vert \cdot \Vert _{{H^{k+1}(\Omega )}}$.

In our error estimates, we will frequently use the following linear operator that maps from $H^{1}(\Omega )\times H^{1}(\Omega )$ onto ${\mathcal {V}}_h^k\times {\mathcal {V}}_h^k$,

$$\begin{aligned} \Pi _h \begin{pmatrix} \varphi \\ \psi \\ \end{pmatrix} := \begin{pmatrix} \Pi _h^1(\varphi ,\psi ,\beta _1)\\ \Pi _h^2(\psi ,\varphi ,\beta _2)\\ \end{pmatrix} = \begin{pmatrix} P_h^+\left( (\frac{1}{2}+\alpha )\varphi +\beta _1\psi \right) +P_h^-\left( (\frac{1}{2}-\alpha )\varphi -\beta _1\psi \right) \\ P_h^+\left( (\frac{1}{2}-\alpha )\psi +\beta _2\varphi \right) +P_h^-\left( (\frac{1}{2}+\alpha )\psi -\beta _2\varphi \right) \end{pmatrix}. \end{aligned}$$

(2.9)

One would want to keep in mind that the operator $\Pi _h$ should have been written as $\Pi _{h}^{\alpha , \beta _1, \beta _2}$ and we omit the parameter dependence for brevity. Associated with the operator $\Pi _h$, we define $(\eta _\varphi , \eta _\psi )^\text {T}$ as

$$\begin{aligned} \begin{pmatrix} \eta _\varphi \\ \eta _\psi \\ \end{pmatrix} := \begin{pmatrix} \varphi \\ \psi \\ \end{pmatrix} -\Pi _h \begin{pmatrix} \varphi \\ \psi \\ \end{pmatrix} = \begin{pmatrix} \varphi -\Pi _h^1 (\varphi ,\psi ,\beta _1) \\ \psi -\Pi _h^2 (\psi ,\varphi ,\beta _2)\\ \end{pmatrix}. \end{aligned}$$

(2.10)

Let $\varphi _h, \psi _h\in {\mathcal {V}}_h^k$ be some approximations for $\varphi$ and $\psi$, respectively, we will also use the following notation in our analysis

$$\begin{aligned} \begin{pmatrix} \zeta _\varphi \\ \zeta _\psi \\ \end{pmatrix} := \Pi _h \begin{pmatrix} \varphi \\ \psi \\ \end{pmatrix} -\begin{pmatrix} \varphi _h\\ \psi _h\\ \end{pmatrix} = \begin{pmatrix} \Pi _h^1 (\varphi ,\psi ,\beta _1)-\varphi _h \\ \Pi _h^2 (\psi ,\varphi ,\beta _2)-\psi _h\\ \end{pmatrix}. \end{aligned}$$

(2.11)

Note that the following decomposition of the errors $e_\varphi =\varphi -\varphi _h$ and $e_\psi =\psi -\psi _h$ holds

$$\begin{aligned} e_\varphi =\eta _\varphi +\zeta _\varphi , \quad e_\psi =\eta _\psi +\zeta _\psi . \end{aligned}$$

(2.12)

The operator $\Pi _h$ was first introduced in [9], and its main properties are summarized in the following lemma.

Lemma 2.1

(Lemma 2.4 in [9]). Consider $(\varphi ,\psi ) \in H^{k+1}(\Omega )\times H^{k+1}(\Omega )$. For any given $\alpha$, $\beta _1$, $\beta _2$, the operator $\Pi _h$ has the following properties:

$$\begin{aligned} (i)&\quad \int _{I_j} \eta _\varphi v_xdx=0, \quad \int _{I_j}\eta _\psi w_xdx=0, \quad \forall v,w\in {\mathcal {V}}_h^k, \quad \forall j, \end{aligned}$$

(2.13)

$$\begin{aligned} (ii)&\quad \left\| \begin{pmatrix} \varphi \\ \psi \\ \end{pmatrix} {-}\Pi _h \begin{pmatrix} \varphi \\ \psi \\ \end{pmatrix} \right\| {\le } C_*\left( 1{+}|\alpha |{+}\max (|\beta _1|,|\beta _2|)\right) h^{k+1}\left( \Vert \psi \Vert _{H^{k+1}(\Omega )}{+}\Vert \varphi \Vert _{H^{k+1}(\Omega )}\right) . \end{aligned}$$

(2.14)

If we further assume $\alpha ^2+\beta _1\beta _2=\frac{1}{4}$, we have

$$\begin{aligned} (iii)&\quad \Pi _h \text{ defines } \text{ a } \text{ projection, } \text{ that } \text{ is } (\Pi _h)^2=\Pi _h, \end{aligned}$$

(2.15)

$$\begin{aligned} (iv)&\quad F_1(\eta _\varphi ,\eta _\psi ,\beta _1)_{j-\frac{1}{2}}=0,\quad F_2(\eta _\psi ,\eta _\varphi ,\beta _2)_{j-\frac{1}{2}}=0, \quad \forall j. \end{aligned}$$

(2.16)

3 DG Methods for Even Order Equations

In this section, we consider even order equations, which include the heat equation (1.1), a fourth order equation (1.2), and the arbitrary even order equations (1.3). The boundary conditions are periodic. For each equation, we will start with its first order form, and apply a general DG formulation. The numerical fluxes are given as the linear combinations of averages values of fluxes, jumps of the solution and the auxiliary variables at cell interfaces, and they involve several parameters. We then identify the conditions on these parameters, such that the DG methods will be $L^2$ stable; the parameters are further specified in order for the solution and some of auxiliary variables to be optimally accurate in the $L^2$ norm. To prove the optimality of the schemes analytically, one or more than one projection operator will be designed, tailored for each choice of the numerical fluxes, in order to eliminate those terms at cell interfaces that would otherwise contribute to the sub-optimality of the error estimates. We want to point out that Sects. 4 and 5 follow a similar structure in the presentation.

3.1 DG Methods for the Heat Equation

In this subsection, we will formulate and analyze DG methods for the heat equation (1.1). Start with the first order form of the equation,

$$\begin{aligned} u_t - p_x=0, \quad p- u_x=0. \end{aligned}$$

(3.1)

A general DG method can be given as follows. Look for $u_h$, $p_h$$\in {\mathcal {V}}_h^k$ such that for any $v$, $w$$\in {\mathcal {V}}_h^k$, and for any j,

$$\begin{aligned} \left( (u_h)_t,v\right) {_{I_j}}+ (p_h,v_x){_{I_j}}- \left( F_p v^-\right) _{{j+\frac{1}{2}}}+\left( F_pv^+\right) _{{j-\frac{1}{2}}}&=0, \end{aligned}$$

(3.2)

$$\begin{aligned} \left( p_h,w\right) {_{I_j}}+ (u_h,w_x){_{I_j}}- \left( F_u w^-\right) _{{j+\frac{1}{2}}} +\left( F_uw^+\right) _{{j-\frac{1}{2}}}&=0. \end{aligned}$$

(3.3)

Here, $F_p$ and $F_u$ in (3.2)–(3.3) are numerical fluxes, which are single-valued functions defined on the cell interfaces and should be designed to ensure the numerical stability and accuracy of numerical solutions. We here consider a family of numerical fluxes, namely,

$$\begin{aligned} F_p=F_1(p_h,u_h,\beta _1), \quad F_u=F_2(u_h,p_h,\beta _2), \end{aligned}$$

(3.4)

and they will correspondingly define a family of DG methods. Note that the numerical fluxes (3.4) include some commonly used ones, such as the central fluxes with $\alpha =\beta _1=\beta _2=0$, and the alternating fluxes with $\alpha =\pm \frac{1}{2}$, $\beta _1=\beta _2=0$. When $\beta _2=0$, the auxiliary variable $p_h$ could be locally expressed in terms of $u_h$, and the DG methods (3.2)–(3.4) become LDG methods.

With the periodic boundary conditions, we sum up the scheme (3.2)–(3.3) over j and reach a more compact form of the scheme: look for $u_h,p_h\in {\mathcal {V}}_h^k$ such that

$$\begin{aligned} B(u_h,p_h;v,w)=0, \quad \forall v,w\in {\mathcal {V}}_h^k, \end{aligned}$$

(3.5)

where

$$\begin{aligned} B(u_h,p_h;v,w)&=\int _{\Omega }(u_h)_tvdx+\sum \limits _{j}\left( \int _{I_j}p_hv_x dx+ (F_p[v])_{j-\frac{1}{2}}\right) \nonumber \\&\quad +\int _{\Omega }p_hwdx +\sum \limits _{j}\left( \int _{I_j}u_hw_x dx+(F_u[w])_{j-\frac{1}{2}}\right) . \end{aligned}$$

(3.6)

3.1.1 $L^2$ Stability

In this subsection, the $L^2$ stability is established for the semi-discrete DG method (3.2)–(3.3) with the general numerical fluxes (3.4). The conditions on the parameters $\alpha , \beta _1, \beta _2$ in the numerical fluxes are identified to ensure the stability.

Theorem 3.1

With $\beta _1\geqslant 0, \ \beta _2\geqslant 0$, the semi-discrete DG scheme (3.2)–(3.3) (or (3.5)) with the numerical fluxes (3.4) satisfies

$$\begin{aligned} \Vert u_h(T)\Vert ^2_\Omega +2\int _0^T \Vert p_h\Vert ^2_\Omega dt\le \Vert u_h(0)\Vert ^2_\Omega , \end{aligned}$$

(3.7)

where T is the final time.

Proof

We start with introducing $H(\varphi ,\psi )=\sum \limits _{j}H_j(\varphi ,\psi )$, where

$$\begin{aligned} H_j(\varphi ,\psi )&=\int _{I_j}\psi \varphi _xdx-(F_1(\psi ,\varphi , \beta _1)\varphi ^-)_{j+\frac{1}{2}}+(F_1(\psi ,\varphi , \beta _1)\varphi ^+)_{j-\frac{1}{2}}\nonumber \\&\quad +\int _{I_j}\varphi \psi _xdx-(F_2(\varphi ,\psi , \beta _2)\psi ^-)_{j+\frac{1}{2}}+(F_2(\varphi ,\psi , \beta _2)\psi ^+)_{j-\frac{1}{2}}. \end{aligned}$$

(3.8)

Using the definitions of the fluxes $F_1, F_2$ in (2.2)–(2.3), we have

$$\begin{aligned} H(\varphi ,\psi )=\sum \limits _{j}\left( -[\varphi \psi ]+F_1[\varphi ]+F_2[\psi ]\right) _{j-\frac{1}{2}}=\sum \limits _{j}\left( \beta _1[\varphi ]^2+\beta _2[\psi ]^2\right) _{j-\frac{1}{2}}. \end{aligned}$$

(3.9)

We now take the test functions $v=u_h$, $w=p_h$ in (3.5), and with the definition of numerical fluxes $F_p, F_u$ in (3.4), we get

$$\begin{aligned} B(u_h,p_h;u_h,p_h)&=\frac{1}{2}\frac{d}{dt}\int _\Omega u_h^2dx+\int _\Omega p_h ^2dx+H(u_h, p_h)\nonumber \\&=\frac{1}{2}\frac{d}{dt}\int _\Omega u_h^2dx+\int _\Omega p_h ^2dx+ \sum \limits _{j}\left( \beta _1[u_h]^2+\beta _2[p_h]^2\right) _{j-\frac{1}{2}}=0. \end{aligned}$$

(3.10)

Finally under the conditions $\beta _1\geqslant 0$ and $\beta _2\geqslant 0$, we reach the energy stability relation,

$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\int _\Omega u_h^2dx+\int _\Omega p_h ^2dx=-\sum \limits _{j}\left( \beta _1[u_h]^2+\beta _2[p_h]^2\right) _{j-\frac{1}{2}}\le 0. \end{aligned}$$

(3.11)

Integrating (3.11) over [0, T], we get (3.7) about stability for Eq. (1.1). $\square$

3.1.2 $L^2$ Error Estimates

In this subsection, we will establish that the DG methods (3.2)–(3.3) with a sub-family of the numerical fluxes (3.4) are optimally accurate in the $L^2$ norm when the exact solution is sufficiently smooth. The analysis is based on the energy relation in Theorem 3.1, approximation properties of the discrete space ${\mathcal {V}}_h^k$, and a special choice of a projection operator.

Theorem 3.2

For the semi-discrete DG scheme (3.2)–(3.3) with the numerical fluxes (3.4) where the parameters satisfy $\alpha ^2+\beta _1\beta _2=\frac{1}{4}$ and $\beta _i\ge 0$, $i=1,2$, the following error estimates hold when the exact solution u is sufficiently smooth,

$$\begin{aligned} \Vert u-u_h\Vert ^2_\Omega \le Ch^{2k+2},\quad \int _0^T\Vert p-p_h\Vert ^2_\Omega dt\le Ch^{2k+2}. \end{aligned}$$

(3.12)

Here $p=u_x$, and the constant C depends on k, the final time T, $\Vert u\Vert _{L^\infty ((0,T);H^{k+2}(\Omega ))}$ and $\Vert u_t\Vert _{L^\infty ((0,T);H^{k+1}(\Omega ))}$ but not on h.

Proof

Since the numerical fluxes (3.4) are consistent, the exact solution u of the heat equation (1.1) and $p=u_x$ satisfy

$$\begin{aligned} B(u,p;v,w)=0, \quad \forall v,w\in {\mathcal {V}}_h^k, \end{aligned}$$

(3.13)

hence we get the error equation

$$\begin{aligned} B(e_u,e_p;v,w)=0,\quad \forall v,w\in {\mathcal {V}}_h^k, \end{aligned}$$

(3.14)

where $e_p=p-p_h$ and $e_u=u-u_h$. By using the following projection,

$$\begin{aligned} \Pi _h \begin{pmatrix} p\\ u\\ \end{pmatrix} = \begin{pmatrix} \Pi _h^1 (p,u,\beta _1) \\ \Pi _h^2 (u,p,\beta _2)\\ \end{pmatrix}, \end{aligned}$$

(3.15)

we can decompose the errors $e_p$ and $e_u$ into $e_p=\eta _p+\zeta _p$ and $e_u=\eta _u+\zeta _u$ based on (2.10)–(2.11). With the linearity of B, the error equation (3.14) becomes

$$\begin{aligned} B(\zeta _u,\zeta _p;v,w)=-B(\eta _u,\eta _p;v,w), \quad \forall v,w\in {\mathcal {V}}_h^k. \end{aligned}$$

(3.16)

We now take $v=\zeta _u$, $w=\zeta _p$ in (3.16). Following the similar derivation to get (3.10) and the definition of B in (3.6), we have

$$\begin{aligned} B(\zeta _u,\zeta _p;\zeta _u,\zeta _p)&=\frac{1}{2}\frac{d}{dt}\int _\Omega \zeta _u^2dx+\int _\Omega \zeta _p ^2dx+\sum \limits _{j}\left( \beta _1[\zeta _u]^2+\beta _2[\zeta _p]^2\right) _{j-\frac{1}{2}}, \end{aligned}$$

(3.17)

$$\begin{aligned} B(\eta _u,\eta _p;\zeta _u,\zeta _p)&=\int _\Omega (\eta _u)_t\zeta _udx+\int _\Omega \eta _p \zeta _p dx +\sum \limits _{j}\int _{I_j}(\eta _p(\zeta _u)_x+\eta _u(\zeta _p)_x)dx\nonumber \\&\quad + \sum \limits _{j}\left( F_1(\eta _p,\eta _u, \beta _1)[\zeta _u]+F_2(\eta _u,\eta _p, \beta _2)[\zeta _p]\right) _{j-\frac{1}{2}}. \end{aligned}$$

(3.18)

Under the assumption $\alpha ^2+\beta _1\beta _2=\frac{1}{4}$, we can use the properties of $\Pi _h$ in Lemma 2.1 and get

$$\begin{aligned} \int _{I_j}\eta _p(\zeta _u)_xdx=0, \int _{I_j}\eta _u(\zeta _p)_xdx=0, \quad F_1(\eta _p,\eta _u, \beta _1)=0,F_2(\eta _u,\eta _p, \beta _2)=0. \end{aligned}$$

(3.19)

Combining (3.16)–(3.19), we obtain

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt}\int _\Omega \zeta _u^2dx+\int _\Omega \zeta _p ^2dx+\sum \limits _{j}\left( \beta _1[\zeta _u]^2+\beta _2[\zeta _p]^2\right) _{j-\frac{1}{2}}=-\int _\Omega (\eta _u)_t\zeta _udx-\int _\Omega \eta _p \zeta _p dx\nonumber \\&\quad \le \frac{1}{2} \int _\Omega (2(\eta _u)_t^2+(\eta _p)^2)dx+ \frac{1}{4}\int _\Omega (\zeta _u^2+2\zeta _p^2)dx. \end{aligned}$$

(3.20)

Based on the approximation property (2.14) of $\Pi _h$ and $(\eta _u)_t=\eta _{u_t}$, we know $\Vert (\eta _u)_t\Vert \le Ch^{k+1}$ and $\Vert \eta _p\Vert \le Ch^{k+1}$. Therefore,

$$\begin{aligned}&\frac{d}{dt}\int _\Omega \zeta _u^2dx+\int _\Omega \zeta _p^2dx+2\sum \limits _{j}\left( \beta _1[\zeta _u]^2+\beta _2[\zeta _p]^2\right) _{j-\frac{1}{2}}\le Ch^{2k+2}+\frac{1}{2}\int _\Omega \zeta _u^2dx. \end{aligned}$$

(3.21)

Note that we use the initialization $u_h(x,0)=P_h u(x,0)$ which can be bounded by

$$\begin{aligned} \left\| \Pi _h^2(u,p,\beta _2) - u_h\right\| _{t=0} =\left\| \Pi _h^2(u,p,\beta _2)-u+u-P_hu \right\| _{t=0}\le Ch^{k+1}. \end{aligned}$$

(3.22)

Now we can apply the Gronwall’s inequality and obtain

$$\begin{aligned} \Vert \zeta _u(\cdot ,t)\Vert ^2_\Omega \le Ch^{2k+2}, \quad \int _0^T \Vert \zeta _p\Vert ^2_\Omega dt\le Ch^{2k+2}. \end{aligned}$$

Finally, using the approximation results in (2.14), we reach

$$\begin{aligned}&\Vert u-u_h\Vert ^2_\Omega \le 2(\Vert \zeta _u\Vert ^2_\Omega +\Vert \eta _u\Vert ^2_\Omega )\le Ch^{2k+2},\\&\int _0^T\Vert p-p_h\Vert ^2_\Omega dt\le \int _0^T 2(\Vert \zeta _p\Vert ^2_\Omega +\Vert \eta _p\Vert ^2_\Omega )dt\le Ch^{2k+2}. \end{aligned}$$

Through the proof, the constant C depends on k, the finial time T, $\Vert u\Vert _{L^\infty ((0,T);H^{k+2}(\Omega ))}$ and $\Vert u_t\Vert _{L^\infty ((0,T);H^{k+1}(\Omega ))}$ but not on h. $\square$

Remark 3.3

Similar as in [9], the DG methods with the more general fluxes

$$\begin{aligned} F_p=\{p_h\}+\alpha _1[p_h]+\beta _1[u_h],\quad F_u=\{u_h\}+\alpha _2[u_h]+\beta _2[p_h], \end{aligned}$$

where $\beta _i\ge 0,i=1,2$ and $(\alpha _1+\alpha _2)^2\le 4\beta _1\beta _2$, also have the energy stability (3.7). Such DG methods however are often sub-optimal in their accuracy.

Remark 3.4

Just as in Theorem 2.6 of [9], the condition $\alpha ^2+\beta _1\beta _2=\frac{1}{4}$ on the parameters in the numerical fluxes can be further relaxed in order for the proposed schemes to be $L^2$ optimal when solving the heat equation. For instance, we can require

$$\begin{aligned} \alpha ^2+\beta _1\beta _2=\frac{1}{4}+ch^{\delta }, \quad \delta \ge \frac{1}{2},\quad \min (\beta _1,\beta _2)> 0, \end{aligned}$$

where c is a constant independent of h.

3.2 DG Methods for a Fourth Order Equation

In this subsection, we will formulate and analyze DG methods for the fourth order equation (1.2). We start with rewriting the equation into its first order system,

$$\begin{aligned} u_t+p_x=0,\quad p=q_x,\quad q=r_x,\quad r=u_x, \end{aligned}$$

(3.23)

and then apply a general DG method. That is, to look for $u_h,p_h,q_h,r_h\in {\mathcal {V}}_h^k$ such that for any $v,w,z,g\in {\mathcal {V}}_h^k$ and for any j

$$\begin{aligned} \left( (u_h)_t,v\right) {_{I_j}}- (p_h,v_x){_{I_j}}+ \left( F_p v^-\right) _{{j+\frac{1}{2}}}-\left( F_pv^+\right) _{{j-\frac{1}{2}}}&=0, \end{aligned}$$

(3.24)

$$\begin{aligned} \left( p_h,w\right) {_{I_j}}+ (q_h,w_x){_{I_j}}- \left( F_q w^-\right) _{{j+\frac{1}{2}}} +\left( F_qw^+\right) _{{j-\frac{1}{2}}}&=0, \end{aligned}$$

(3.25)

$$\begin{aligned} \left( q_h,z\right) {_{I_j}}\ + (r_h,z_x){_{I_j}}- \left( F_r z^-\right) _{{j+\frac{1}{2}}} +\left( F_rz^+\right) _{{j-\frac{1}{2}}}&=0, \end{aligned}$$

(3.26)

$$\begin{aligned} \left( r_h,g\right) {_{I_j}}+ (u_h,g_x){_{I_j}}- \left( F_u g^-\right) _{{j+\frac{1}{2}}} +\left( F_ug^+\right) _{{j-\frac{1}{2}}}&=0. \end{aligned}$$

(3.27)

The terms $F_p$, $F_q$, $F_r$ and $F_u$ are numerical fluxes. They are defined as linear combinations of the averages of fluxes and the jumps of the unknown solutions $u_h$ and $p_h$, $q_h$ and $r_h$, and are chosen as

$$\begin{aligned} F_p{=}F_1(p_h,u_h,-\beta _1),\;F_q=F_2(q_h,r_h,\beta _2),\;F_r=F_1(r_h,q_h,\beta _1),\;F_u=F_2(u_h,p_h,-\beta _2). \end{aligned}$$

(3.28)

In order for simplifying the conditions and flexibly extending to general even-order equations, the parameter $\alpha$ is the same for all $F_p,F_u, F_r,F_q$, while $\beta _1, \beta _2$ in $F_p,F_u$ are related to those in $F_r,F_q$.

With the periodic boundary conditions, we sum up all the equations in (3.24)–(3.27) over j and obtain a more compact form of the scheme: look for $u_h,p_h,q_h,r_h\in {\mathcal {V}}_h^k$ such that

$$\begin{aligned} B(u_h,p_h,q_h,r_h;v,w,z,g)=0, \quad \forall v,w,z,g\in {\mathcal {V}}_h^k, \end{aligned}$$

(3.29)

where

$$\begin{aligned} B(u_h,p_h,q_h,r_h;v,w,z,g)&=\int _{\Omega }\left( (u_h)_tv+p_hw+ q_hz+r_hg\right) dx\nonumber \\&\quad + \sum \limits _{j}\int _{I_j}\left( -p_hv_x+q_hw_x+r_hz_x+ u_hg_x\right) dx\nonumber \\&\quad +\sum \limits _{j}\left( -F_p[v]+F_q[w]+F_r[z]+F_u[g]\right) _{j-\frac{1}{2}}. \end{aligned}$$

(3.30)

3.2.1 $L^2$ Stability

In this subsection, the $L^2$ stability is established for the semi-discrete DG method (3.24)–(3.27) with the general numerical fluxes (3.28). Just as in [20], in order for the proposed methods to be optimally accurate, more than one energy equation is needed in the presence of multiple auxiliary unknowns, also see Remark 3.6.

$\bullet$The first energy equation By taking the test functions $v=u_h$, $w=r_h$, $z=q_h$ and $g=-p_h$ in (3.29), we obtain

$$\begin{aligned} 0&=B(u_h,p_h,q_h,r_h;u_h,r_h,q_h,-p_h)=\int _{\Omega }\left( (u_h)_tu_h+q_h^2\right) dx\nonumber \\&\quad +\sum \limits _{j}\left( [u_hp_h]-[q_hr_h]-F_p[u_h]+F_q[r_h]+F_r[q_h]-F_u[p_h]\right) _{j-\frac{1}{2}}. \end{aligned}$$

(3.31)

Using the definitions of $F_p,F_q,F_r,F_u$ in (3.28), we get

$$\begin{aligned} 0&=B(u_h,p_h,q_h,r_h;u_h,r_h,q_h,-p_h)=\int _{\Omega }\left( (u_h)_tu_h+q_h^2\right) dx\nonumber \\&\quad +\sum \limits _{j}\left( \beta _1([u_h]^2+[q_h]^2)+\beta _2([p_h]^2+[r_h]^2)\right) _{j-\frac{1}{2}}. \end{aligned}$$

(3.32)

$\bullet$The second energy equation We next take the time derivative for the Eq. (3.27) and sum it up, together with (3.24)–(3.26), over j. By taking the test functions $v=-\frac{1}{2}q_h$, $w=\frac{1}{2}p_h$, $z=\frac{1}{2}(u_h)_t$ and $g=\frac{1}{2}r_h$, we get

$$\begin{aligned} 0&=B\left( u_h,p_h,q_h,(r_h)_t;-\frac{1}{2}q_h,\frac{1}{2}p_h,\frac{1}{2}(u_h)_t,\frac{1}{2}r_h\right) =\frac{1}{2}\int _{\Omega }\left( p_h^2+(r_h)_tr_h\right) dx\nonumber \\&\quad + \frac{1}{2}\sum \limits _{j}\int _{I_j}\left( p_h(q_h)_x+q_h(p_h)_x+r_h(u_h)_{tx}+ (u_h)_t(r_h)_x\right) dx\nonumber \\&\quad +\frac{1}{2}\sum \limits _{j}\left( F_p[q_h]+F_q[p_h]+F_r[(u_h)_t]+(F_u)_t[r_h]\right) _{j-\frac{1}{2}}. \end{aligned}$$

(3.33)

Using the definitions of $F_p$, $F_q$, $F_r$, $F_u$ in (3.28), we have

$$\begin{aligned} 0&=B\left( u_h,p_h,q_h,(r_h)_t;-\frac{1}{2}q_h,\frac{1}{2}p_h,\frac{1}{2}(u_h)_t,\frac{1}{2}r_h\right) =\frac{1}{2}\int _{\Omega }\left( p_h^2+(r_h)_tr_h\right) dx\nonumber \\&\quad +\frac{1}{2}\sum \limits _{j}\left( \beta _1([q_h][(u_h)_t]-[u_h][q_h])+\beta _2([r_h][p_h]-[(p_h)_t][r_h])\right) _{j-\frac{1}{2}}. \end{aligned}$$

(3.34)

$\bullet$The third energy equation Here, we take the time derivative for the Eqs. (3.26)–(3.27) and sum it up, together with (3.24)–(3.25), over j. Then, we take the test functions $v=\frac{1}{2}(u_h)_t$, $w=\frac{1}{2}(r_h)_t$, $z=\frac{1}{2}q_h$ and $g=-\frac{1}{2}p_h$. Using the definitions of (3.29) and $F_p$, $F_q$, $F_r$, $F_u$ in (3.28), we have

$$\begin{aligned} 0&=B\left( u_h,p_h,(q_h)_t,(r_h)_t;\frac{1}{2}(u_h)_t,\frac{1}{2}(r_h)_t,\frac{1}{2}q_h,-\frac{1}{2}p_h\right) =\frac{1}{2}\int _{\Omega }\left( (u_h)_t^2+q_h(q_h)_t\right) dx\nonumber \\&\quad +\frac{1}{2}\sum \limits _{j}\left( [(u_h)_tp_h]{-}[q_h(r_h)_t]{-}F_p[(u_h)_t]+F_q[(r_h)_t]+(F_r)_t[q_h]-(F_u)_t[p_h]\right) _{j-\frac{1}{2}}\nonumber \\&=\frac{1}{2}\int _{\Omega }\left( (u_h)_t^2+(q_h)_tq_h\right) dx\nonumber \\&\quad +\frac{1}{2}\sum \limits _{j}(\beta _1([(q_h)_t][q_h]+[u_h][(u_h)_t])+\beta _2([r_h][(r_h)_t]+[(p_h)_t][p_h]))_{j-\frac{1}{2}}. \end{aligned}$$

(3.35)

$\bullet$The fourth energy equation We take the time derivative for (3.25)–(3.27) and sum them up with (3.24) over j. Using the definition of B in (3.29) and taking the test functions $v=-\frac{1}{2}(q_h)_t$, $w=\frac{1}{2}p_h$, $z=\frac{1}{2}(u_h)_t$ and $g=\frac{1}{2}(r_h)_t$, we obtain

$$\begin{aligned} 0&=B(u_h,(p_h)_t,(q_h)_t,(r_h)_t;-\frac{1}{2}(q_h)_t,\frac{1}{2}p_h,\frac{1}{2}(u_h)_t,\frac{1}{2}(r_h)_t)=\frac{1}{2}\int _{\Omega }\left( (p_h)_tp_h+((r_h)_t)^2\right) dx\nonumber \\&\quad +\frac{1}{2}\sum \limits _{j}\left( -[(u_h)_t(r_h)_t]-[p_h(q_h)_t]+F_p[(q_h)_t]+(F_q)_t[p_h]+(F_r)_t[(u_h)_t]+(F_u)_t[(r_h)_t]\right) _{j-\frac{1}{2}}\nonumber \\&=\frac{1}{2}\int _{\Omega }\left( (p_h)_tp_h+((r_h)_t)^2\right) dx\nonumber \\&\quad +\frac{1}{2}\sum \limits _{j}(\beta _1([(q_h)_t][(u_h)_t]-[(q_h)_t][u_h])+\beta _2([(r_h)_t][p_h]-[(p_h)_t][(r_h)_t])_{j-\frac{1}{2}}. \end{aligned}$$

(3.36)

$\bullet$The fifth energy equation Finally we take the time derivative for Eqs. (3.24)–(3.27), sum them up over j. With the test functions $v=(u_h)_t$, $w=(r_h)_t$, $z=(q_h)_t$ and $g=-(p_h)_t$, we get

$$\begin{aligned} 0&=B((u_h)_t,(p_h)_t,(q_h)_t,(r_h)_t;(u_h)_t,(r_h)_t,(q_h)_t,-(p_h)_t)=\int _{\Omega }\left( (u_h)_{tt}(u_h)_t+(q_h)_t^2\right) dx\nonumber \\&\quad +\sum \limits _{j}\left( \beta _1([(u_h)_t]^2+[(q_h)_t]^2)+\beta _2([(p_h)_t]^2+[(r_h)_t]^2)\right) _{j-\frac{1}{2}}. \end{aligned}$$

(3.37)

Combining (3.32) and (3.34)–(3.37), we have

$$\begin{aligned}&\frac{1}{4}\frac{d}{dt}\int _{\Omega }\left( 2u_h^2{+}p_h^2{+}q_h^2{+}r_h^2{+}2(u_h)_t^2\right) dx {+}\frac{1}{2} \int _{\Omega }\left( 2q_h^2{+}p_h^2{+}(u_h)^2_t{+}(r_h)_t^2{+}2(q_h)_t^2\right) dx\nonumber \\&\quad +\sum _j \Big (\beta _1\Lambda ([u_h], [(u_h)_t], [q_h], [(q_h)_t])+\beta _2\Lambda ([(p_h)_t], [p_h], [r_h], [(r_h)_t])\Big )_{{j-\frac{1}{2}}} =0, \end{aligned}$$

(3.38)

where $\Lambda (a, b, c, d)$ is a non-negative quadratic form, defined as

$$\begin{aligned} \Lambda (a, b, c, d)&=a^2+b^2+c^2+d^2 + \frac{1}{2} ( ab + cd + bc-ac +bd-ad) \ge 0. \end{aligned}$$

(3.39)

If we require $\beta _1\ge 0,\beta _2\ge 0$, then (3.38) gives

$$\begin{aligned}&\frac{1}{4}\frac{d}{dt}\int _{\Omega }\left( 2u_h^2{+}p_h^2{+}q_h^2{+}r_h^2{+}2(u_h)_t^2\right) dx {+} \int _{\Omega }\left( q_h^2{+}\frac{1}{2}p_h^2{+}r_h^2{+}\frac{1}{2}(r_h)_t^2{+}(q_h)^2_t\right) dx\le 0. \end{aligned}$$

(3.40)

This leads to the following theorem.

Theorem 3.5

Using the numerical fluxes (3.28) with $\beta _1\ge 0$ and $\beta _2\ge 0$, the numerical solutions of the semi-discrete DG method (3.29) for the fourth order equation satisfy

$$\begin{aligned}&2\Vert u_h(T)\Vert ^2_\Omega +\Vert p_h(T)\Vert ^2_\Omega +\Vert q_h(T)\Vert ^2_\Omega +\Vert r_h(T)\Vert ^2_\Omega +2\Vert (u_h)_t(T)\Vert ^2_\Omega \\&\quad \le 2\Vert u_h(0)\Vert ^2_\Omega +\Vert p_h(0)\Vert ^2_\Omega +\Vert q_h(0)\Vert ^2_\Omega +\Vert r_h(0)\Vert ^2_\Omega +2\Vert (u_h)_t(0)\Vert ^2_\Omega . \end{aligned}$$

Here T is the final time.

Remark 3.6

Compared with the analysis in [20], more energy equations are needed in our analysis to ensure the optimal accuracy of the proposed methods due to the extra parameters $\beta _1,\beta _2$ and $\alpha$ ($\alpha \ne \pm \frac{1}{2}$). When $\beta _1=\beta _2=0$, only the first and the second energy equations are needed just as in [20].

3.2.2 $L^2$ Error Estimates

In this subsection, we will prove the optimal a priori$L^2$ error estimate for the DG method (3.29) for the fourth order equation when the exact solution is sufficiently smooth, under the following assumption for the parameters in the numerical fluxes (3.28)

$$\begin{aligned} \alpha ^2+\beta _1\beta _2=\frac{1}{4}, \quad \beta _1\ge 0,\; \beta _2\ge 0. \end{aligned}$$

(3.41)

Since the numerical fluxes (3.28) are consistent, the exact solution u, $r=u_x$, $q=r_x$, and $p=q_x$ satisfy

$$\begin{aligned} B(u,p,q,r;v,w,z,g)=0, \quad \forall v,w,z,g\in {\mathcal {V}}_h^k, \end{aligned}$$

(3.42)

hence we get the error equation

$$\begin{aligned} B(e_u,e_p,e_q,e_r;v,w,z,g)=0, \quad \forall v,w,z,g\in {\mathcal {V}}_h^k. \end{aligned}$$

(3.43)

Here $e_\phi =\phi -\phi _h$, with $\phi =u, p, q, r$ are error functions. To ensure the error estimates to be optimal, we use the special projections

$$\begin{aligned} \Pi _h \begin{pmatrix} p\\ u\\ \end{pmatrix} = \begin{pmatrix} \Pi _h^1 (p,u,-\beta _1)\\ \Pi _h^2 (u,p,-\beta _2)\\ \end{pmatrix} ,\quad \Pi _h \begin{pmatrix} r\\ q\\ \end{pmatrix} = \begin{pmatrix} \Pi _h^1 (r,q,\beta _1)\\ \Pi _h^2 (q,r,\beta _2)\\ \end{pmatrix}, \end{aligned}$$

(3.44)

with which the error functions can be decomposed into $e_\phi =\eta _\phi +\zeta _\phi , \phi =u, p, q, r$ based on (2.10)–(2.11), and the error equation becomes

$$\begin{aligned} B(\zeta _u, \zeta _p, \zeta _q, \zeta _r;v,w,z,g)=-B(\eta _u, \eta _p, \eta _q, \eta _r;v,w,z,g),\quad \forall v,w,z,g\in {\mathcal {V}}_h^k.\qquad \end{aligned}$$

(3.45)

We choose to set the initial condition $p_h(x,0)$ as follows,

$$\begin{aligned} p_h(x,0)=P_h^+ p(x,0), \quad p(x,0)=u_{xxx}(x,0). \end{aligned}$$

(3.46)

Using $p_h(x,0)$, we can further define the initial data $q_h(x,0), r_h(x,0), u_h(x, 0) \in {\mathcal {V}}_h^k$, satisfying

$$\begin{aligned}&(p_h,w)_{I_j}+(q_h,w_x)|_{I_j}-(\widehat{q}_hw^-)|_{j+\frac{1}{2}}+(\widehat{q}_hw^+)|_{j-\frac{1}{2}}=0,\; \forall w\in {\mathcal {V}}_h^k,\; \nonumber \\&\quad \text{ with }\;{\widehat{q}}_h=q_h^-(x,0) \;\;\text{ and }\;\; \int _\Omega q_h(x,0)dx=\int _\Omega q(x,0)dx, \end{aligned}$$

(3.47)

$$\begin{aligned}&(q_h,w)_{I_j}+(r_h,w_x)|_{I_j}-({\widehat{r}}_hw^-)|_{j+\frac{1}{2}}+({\widehat{r}}_hw^+)|_{j-\frac{1}{2}}=0,\; \forall w\in {\mathcal {V}}_h^k,\; \nonumber \\&\quad \text{ with }\;{\widehat{r}}_h=r_h^+(x,0) \;\;\text{ and }\;\; \int _\Omega r_h(x,0)dx=\int _\Omega r(x,0)dx, \end{aligned}$$

(3.48)

$$\begin{aligned}&(r_h,w)_{I_j}+(u_h,w_x)|_{I_j}-({\widehat{u}}_hw^-)|_{j+\frac{1}{2}}+({\widehat{u}}_hw^+)|_{j-\frac{1}{2}}=0,\; \forall w\in {\mathcal {V}}_h^k,\; \nonumber \\&\quad \text{ with }\;{\widehat{u}}_h=u_h^-(x,0) \;\;\text{ and }\;\; \int _\Omega u_h(x,0)dx=\int _\Omega u(x,0)dx. \end{aligned}$$

(3.49)

Following a similar analysis for Lemma 5.1 in [23], we can prove that the initial conditions above are well-defined with optimal accuracy. And similar to the analysis for Lemma 2.4 in [30], we have the optimal error estimates about $||u_t-(u_h)_t||$ at $t=0$. The results are summarized next with the proof omitted.

Lemma 3.7

Assuming u(x, 0) is sufficiently smooth, the initial conditions in (3.46)–(3.49) are well defined and satisfy the following estimates

$$\begin{aligned}&||p(x,0)-p_h(x,0)||_\Omega \le Ch^{k+1},\; ||q(x,0)-q_h(x,0)||_\Omega \le Ch^{k+1},\nonumber \\&||u(x,0)-u_h(x,0)||_\Omega \le Ch^{k+1},\;||r(x,0)-r_h(x,0)||_\Omega \le Ch^{k+1},\nonumber \\&||u_t(x,0)-(u_h)_t(x,0)||_\Omega \le Ch^{k+1}. \end{aligned}$$

Here $(u_h)_t(x,0)$ is determined by (3.24) with $F_p=p_h^+(x,0)$ at $t=0$. And C depends on $||u(x,0)||_{H^{k+3}(\Omega )}$, and $r(x,0)=u_x(x,0),q(x,0)=u_{xx}(x,0)$.

Next, we will follow the analysis of energy stability and get five error equations to obtain the error estimates.

$\bullet$The first error equation We start with the error equation (3.45) and take the test functions to be $v=\zeta _u, w=\zeta _r, z=\zeta _q, g=-\zeta _p$, and get

$$\begin{aligned} B(\zeta _u, \zeta _p, \zeta _q, \zeta _r;\zeta _u,\zeta _r,\zeta _q,-\zeta _p)=-B(\eta _u, \eta _p, \eta _q, \eta _r;\zeta _u,\zeta _r,\zeta _q,-\zeta _p). \end{aligned}$$

(3.50)

Following the derivation of the first energy equation (3.32), we have

$$\begin{aligned} B(\zeta _u,\zeta _p,\zeta _q,\zeta _r;\zeta _u,\zeta _r,\zeta _q,-\zeta _p)&=\int _{\Omega }\left( (\zeta _u)_t\zeta _u+\zeta _q^2\right) dx\nonumber \\&\quad +\sum \limits _{j}\left( \beta _1([\zeta _u]^2+[\zeta _q]^2)+\beta _2([\zeta _p]^2+[\zeta _r]^2)\right) _{j-\frac{1}{2}}. \end{aligned}$$

(3.51)

And from (3.30),

$$\begin{aligned}&B(\eta _u,\eta _p,\eta _q,\eta _r;\zeta _u,\zeta _r,\zeta _q,-\zeta _p)=\int _{\Omega }\left( (\eta _u)_t\zeta _u+\eta _p\zeta _r+\eta _q\zeta _q-\eta _r\zeta _p\right) dx\nonumber \\&\qquad +\sum \limits _{j}\int _{I_j}\left( (-\eta _p)(\zeta _u)_x+\eta _q(\zeta _r)_x+\eta _r(\zeta _q)_x-\eta _u(\zeta _p)x\right) dx\nonumber \\&\qquad +\sum \limits _{j}\left( -F_p[\zeta _u]+F_q[\zeta _r]+F_r[\zeta _q]-F_u[\zeta _p]\right) _{j-\frac{1}{2}}\nonumber \\&\quad =\int _{\Omega }\left( (\eta _u)_t\zeta _u+\eta _p\zeta _r+\eta _q\zeta _q-\eta _r\zeta _p\right) dx. \end{aligned}$$

(3.52)

Here, the properties (2.13) and (2.16) of $\Pi _h$ in Lemma 2.1 are used. Combining (3.50)–(3.52), we get

$$\begin{aligned}&\int _{\Omega }\left( (\zeta _u)_t\zeta _u+\zeta _q^2\right) dx+\sum \limits _{j}\left( \beta _1([\zeta _u]^2+[\zeta _q]^2)+\beta _2([\zeta _p]^2+[\zeta _r]^2)\right) _{j-\frac{1}{2}}\nonumber \\&\quad =-\int _{\Omega }\left( (\eta _u)_t\zeta _u+\eta _p\zeta _r+\eta _q\zeta _q-\eta _r\zeta _p\right) dx. \end{aligned}$$

(3.53)

$\bullet$The second error equation Following a similar procedure to get the energy equation (3.34), we have

$$\begin{aligned}&B(\zeta _u,\zeta _p,\zeta _q,(\zeta _r)_t;-\frac{1}{2}\zeta _q,\frac{1}{2}\zeta _p,\frac{1}{2}(\zeta _u)_t,\frac{1}{2}\zeta _r)\nonumber \\&\quad = -B\left( \eta _u,\eta _p,\eta _q,(\eta _r)_t;-\frac{1}{2}\zeta _q,\frac{1}{2}\zeta _p,\frac{1}{2}(\zeta _u)_t,\frac{1}{2}\zeta _r\right) , \end{aligned}$$

and

$$\begin{aligned}&B(\zeta _u,\zeta _p,\zeta _q,(\zeta _r)_t;-\frac{1}{2}\zeta _q,\frac{1}{2}\zeta _p,\frac{1}{2}(\zeta _u)_t,\frac{1}{2}\zeta _r)=\frac{1}{2}\int _{\Omega }\left( \zeta _p^2+(\zeta _r)_t\zeta _r\right) dx\nonumber \\&\quad +\frac{1}{2}\sum \limits _{j}\left( \beta _1([\zeta _q][(\zeta _u)_t]-[\zeta _u][\zeta _q])+\beta _2([\zeta _r][\zeta _p]-[(\zeta _p)_t][\zeta _r])\right) _{j-\frac{1}{2}}. \end{aligned}$$

(3.54)

Using the definition of B in (3.30) and the properties (2.13), (2.16) of $\Pi _h$, we get

$$\begin{aligned}&B\left( \eta _u,\eta _p,\eta _q,(\eta _r)_t;-\frac{1}{2}\zeta _q,\frac{1}{2}\zeta _p,\frac{1}{2}(\zeta _u)_t,\frac{1}{2}\zeta _r\right) \nonumber \\&\quad =\frac{1}{2}\int _{\Omega }\left( -(\eta _u)_t\zeta _q+\eta _p\zeta _p+\eta _q(\zeta _u)_t+(\eta _r)_t\zeta _r\right) dx, \end{aligned}$$

(3.55)

hence

$$\begin{aligned}&\frac{1}{2}\int _{\Omega }\left( \zeta _p^2{+}(\zeta _r)_t\zeta _r\right) dx {+}\frac{1}{2}\sum \limits _{j}\left( \beta _1([\zeta _q][(\zeta _u)_t]{-}[\zeta _u][\zeta _q]){+}\beta _2([\zeta _r][\zeta _p]{-}[(\zeta _p)_t][\zeta _r])\right) _{j-\frac{1}{2}}\nonumber \\&\quad =-\frac{1}{2}\int _{\Omega }\left( -(\eta _u)_t\zeta _q+\eta _p\zeta _p+\eta _q(\zeta _u)_t+(\eta _r)_t\zeta _r\right) dx. \end{aligned}$$

(3.56)

$\bullet$The third error equation Following a similar procedure to get the energy equation (3.35), we have

$$\begin{aligned}&B(\zeta _u,\zeta _p,(\zeta _q)_t,(\zeta _r)_t;\frac{1}{2}(\zeta _u)_t,\frac{1}{2}(\zeta _r)_t,\frac{1}{2}\zeta _q,-\frac{1}{2}\zeta _p)=\frac{1}{2}\int _{\Omega }\left( (\zeta _u)_t^2+\zeta _q(\zeta _q)_t\right) dx\nonumber \\&\quad +\frac{1}{2}\sum \limits _{j}(\beta _1([(\zeta _q)_t][\zeta _q]+[\zeta _u][(\zeta _u)_t])+\beta _2([\zeta _r][(\zeta _r)_t]+[(\zeta _p)_t][\zeta _p]))_{j-\frac{1}{2}}. \end{aligned}$$

(3.57)

Using the definition of (3.30) and the properties (2.13), (2.16) of $\Pi _h$, we obtain

$$\begin{aligned}&B\left( \eta _u,\eta _p,(\eta _q)_t,(\eta _r)_t;\frac{1}{2}(\zeta _u)_t,\frac{1}{2}(\zeta _r)_t,\frac{1}{2}\zeta _q,-\frac{1}{2}\zeta _p\right) \\&\quad =\frac{1}{2} \int _{\Omega }\left( (\eta _u)_t(\zeta _u)_t+\eta _p(\zeta _r)_t+(\eta _q)_t\zeta _q-(\eta _r)_t\zeta _p\right) dx, \end{aligned}$$

therefore

$$\begin{aligned}&\frac{1}{2}\int _{\Omega }\left( (\zeta _q)_t\zeta _q+(\zeta _u)_t^2\right) dx+\frac{1}{2}\sum \limits _{j}(\beta _1([(\zeta _q)_t][\zeta _q]+[\zeta _u][(\zeta _u)_t])+\beta _2([\zeta _r][(\zeta _r)_t]+[(\zeta _p)_t][\zeta _p]))_{j-\frac{1}{2}}\nonumber \\&\quad =-\frac{1}{2}\int _{\Omega }\left( (\eta _u)_t(\zeta _u)_t+\eta _p(\zeta _r)_t+(\eta _q)_t\zeta _q-(\eta _r)_t\zeta _p\right) dx. \end{aligned}$$

(3.58)

$\bullet$The fourth error equation Here, we follow the procedure to derive (3.36) and have

$$\begin{aligned}&B\left( \zeta _u,(\zeta _p)_t,(\zeta _q)_t,(\zeta _r)_t;-\frac{1}{2}(\zeta _q)_t,\frac{1}{2}\zeta _p,\frac{1}{2}(\zeta _u)_t,\frac{1}{2}(\zeta _r)_t\right) =\frac{1}{2}\int _{\Omega }\left( (\zeta _p)_t\zeta _p+((\zeta _r)_t)^2\right) dx\nonumber \\&\quad +\frac{1}{2}\sum \limits _{j}\left( \beta _1([(\zeta _q)_t][(\zeta _u)_t]-[\zeta _u][(\zeta _q)_t])+\beta _2([(\zeta _r)_t][\zeta _p]-[(\zeta _p)_t][(\zeta _r)_t])\right) _{j-\frac{1}{2}}. \end{aligned}$$

(3.59)

And from (3.30) and the properties (2.13), (2.16) of $\Pi _h$, we get

$$\begin{aligned}&B\left( \eta _u,(\eta _p)_t,(\eta _q)_t,(\eta _r)_t;-\frac{1}{2}(\zeta _q)_t,\frac{1}{2}\zeta _p,\frac{1}{2}(\zeta _u)_t,\frac{1}{2}(\zeta _r)_t\right) \nonumber \\&\quad =\frac{1}{2}\int _{\Omega }\left( -(\eta _u)_t(\zeta _q)_t+(\eta _p)_t\zeta _p+(\eta _q)_t(\zeta _u)_t+(\eta _r)_t(\zeta _r)_t\right) dx, \end{aligned}$$

(3.60)

therefore

$$\begin{aligned}&\frac{1}{2}\int _{\Omega }\left( (\zeta _p)_t\zeta _p+((\zeta _r)_t)^2\right) dx+\frac{1}{2}\sum \limits _{j}\left( \beta _1([(\zeta _q)_t][(\zeta _u)_t]-[\zeta _u][(\zeta _q)_t]) \right. \nonumber \\&\qquad \left. +\;\beta _2([(\zeta _r)_t][\zeta _p]-[(\zeta _p)_t][(\zeta _r)_t])\right) _{j-\frac{1}{2}}\nonumber \\&\quad =-\frac{1}{2}\int _{\Omega }\left( -(\eta _u)_t(\zeta _q)_t+(\eta _p)_t\zeta _p+(\eta _q)_t(\zeta _u)_t+(\eta _r)_t(\zeta _r)_t\right) dx. \end{aligned}$$

(3.61)

$\bullet$The fifth error equation Last, we follow the procedure to get the fifth energy equation (3.37) and have

$$\begin{aligned}&B((\zeta _u)_t,(\zeta _p)_t,(\zeta _q)_t,(\zeta _r)_t;(\zeta _u)_t,(\zeta _r)_t,(\zeta _q)_t,-(\zeta _p)_t))=\int _{\Omega }\left( (\zeta _u)_{tt}(\zeta _u)_t+(\zeta _q)_t^2\right) dx\nonumber \\&\quad +\sum \limits _{j}\left( \beta _1([(\zeta _q)_t]^2+[(\zeta _u)_t]^2)+\beta _2([(\zeta _p)_t]^2+[(\zeta _r)_t]^2)\right) _{j-\frac{1}{2}}. \end{aligned}$$

(3.62)

Combining the Eq. (3.30) and properties (2.13), (2.16) of $\Pi _h$, we obtain

$$\begin{aligned}&B((\eta _u)_t,(\eta _p)_t,(\eta _q)_t,(\eta _r)_t;(\zeta _u)_t,(\zeta _r)_t,(\zeta _q)_t,-(\zeta _p)_t))\nonumber \\&\quad =\int _{\Omega }\left( (\eta _u)_{tt}(\zeta _u)_t+(\eta _p)_t(\zeta _r)_t+(\eta _q)_t(\zeta _q)_t-(\eta _r)_t(\zeta _p)_t\right) dx. \end{aligned}$$

(3.63)

Thus, we have

$$\begin{aligned}&\int _{\Omega }\left( (\zeta _u)_{tt}(\zeta _u)_t+(\zeta _q)_t^2\right) dx+\sum \limits _{j}\left( \beta _1([(\zeta _q)_t]^2+[(\zeta _u)_t]^2)+\beta _2([(\zeta _p)_t]^2+[(\zeta _r)_t]^2)\right) _{j-\frac{1}{2}}\nonumber \\&\quad =-\int _{\Omega }\left( (\eta _u)_{tt}(\zeta _u)_t+(\eta _p)_t(\zeta _r)_t+(\eta _q)_t(\zeta _q)_t-(\eta _r)_t(\zeta _p)_t\right) dx. \end{aligned}$$

(3.64)

We sum up (3.53), (3.56), (3.58), (3.61) and (3.64), and get

$$\begin{aligned}&\frac{1}{4}\frac{d}{dt}\int _{\Omega }\left( 2\zeta _u^2+\zeta _p^2+\zeta _q^2+\zeta _r^2+2(\zeta _u)_t^2\right) dx + \frac{1}{2}\int _{\Omega }\left( 2\zeta _q^2+\zeta _p^2+(\zeta _u)_t^2+(\zeta _r)_t^2+2(\zeta _q)^2_t\right) dx\nonumber \\&\qquad +\sum _j \Big (\beta _1\Lambda ([\zeta _u], [(\zeta _u)_t], [\zeta _q], [(\zeta _q)_t])+\beta _2\Lambda ([(\zeta _p)_t], [\zeta _p], [\zeta _r], [(\zeta _r)_t])\Big )_{{j-\frac{1}{2}}}\nonumber \\&\quad =P_1+P_2+Q_1+Q_2+R_1+R_2+T_1+T_2. \end{aligned}$$

(3.65)

Here $\Lambda (a, b, c, d)$ is a non-negative quadratic form as in (3.39), while $P_1,P_2,Q_1,Q_2,R_1,R_2,T_1,T_2$ are defined and bounded as below,

$$\begin{aligned} P_1&=-\int _{\Omega }\left( \frac{1}{2}\eta _p-\eta _r+\frac{1}{2}\left( \eta _p\right) _t-\frac{1}{2}\left( \eta _r\right) _t\right) \zeta _pdx\le Ch^{2k+2}+\frac{1}{4}\int _{\Omega }\zeta _p^2dx,\\P_2&=\int _{\Omega }\left( \eta _r\right) _t\left( \zeta _p\right) _t dx,\\ Q_1&=-\int _{\Omega }\left( \eta _q-\frac{1}{2}\left( \eta _u\right) _t+\frac{1}{ 2}\left( \eta _q\right) _t\right) \zeta _qdx\le Ch^{2k+2}+\frac{5}{4}\int _{\Omega }\zeta _q^2dx,\\ Q_2&=-\int _{\Omega }\left( \left( \eta _q\right) _t-\frac{1}{2}\left( \eta _u\right) _t\right) \left( \zeta _q\right) _tdx\le Ch^{2k+2}+\int _{\Omega }\left( \zeta _q\right) _t^2dx,\\ R_1&=-\int _{\Omega }\left( \left( \eta _p+\frac{1}{2}\left( \eta _r\right) _t\right) \zeta _r\right) dx\le Ch^{2k+2}+\frac{1}{4}\int _{\Omega }\zeta _r^2dx,\\ R_2&=-\int _{\Omega }\left( \frac{1}{2}\left( \eta _r\right) _t+\frac{1}{2}\eta _p+\left( \eta _p\right) _t\right) \left( \zeta _r\right) _tdx\le Ch^{2k+2}+\frac{1}{2}\int _{\Omega }\left( \zeta _r\right) _t^2dx,\\ T_1&=-\int _{\Omega }\left( \eta _u\right) _t\zeta _udx\le Ch^{2k+2}+\frac{1}{2}\int _{\Omega }\zeta _u^2dx,\\ T_2&=-\frac{1}{2}\int _{\Omega }\left( \eta _q+\left( \eta _q\right) _t+\left( \eta _u\right) _t+2\left( \eta _u\right) _{tt}\right) \left( \zeta _u\right) _tdx\le Ch^{2k+2}+\int _{\Omega }\left( \zeta _u\right) _t^2dx. \end{aligned}$$

With $\beta _1\ge 0,\beta _2\ge 0$ and the inequalities above, we get

$$\begin{aligned}&\frac{1}{4}\frac{d}{dt}\int _{\Omega }\left( 2\zeta _u^2+\zeta _p^2+\zeta _q^2+\zeta _r^2+2(\zeta _u)_t^2\right) dx\nonumber \\&\quad \le Ch^{2k+2}+\frac{1}{4}\int _{\Omega }(2\zeta _u^2{+}\zeta _p^2+\zeta _q^2+\zeta _r^2+2(\zeta _u)_t^2)dx+P_2. \end{aligned}$$

(3.66)

For $P_2$, we have

$$\begin{aligned} \int ^T_0P_2dt=&\int _{\Omega }(\eta _r)_{t}\zeta _pdx\mid _{0}^{T}-\int ^T_0\int _{\Omega }(\eta _r)_{tt}\zeta _pdxdt\le Ch^{2k+2}+\frac{1}{8}\Vert \zeta _p\Vert ^2_\Omega (T)\nonumber \\ {}&+\frac{3}{8}\int ^T_0\int _{\Omega }\zeta _p^2dxdt. \end{aligned}$$

Integrating in time for (3.66) over [0, T], we obtain

$$\begin{aligned}&\frac{1}{4}\int _{\Omega }\left( 2\zeta _u^2+\frac{1}{2}\zeta _p^2+\zeta _q^2+\zeta _r^2+2(\zeta _u)_t^2\right) dx|_{t=T} \nonumber \\&\quad \le Ch^{2k+2}+ \frac{1}{4}\int _0^T\int _{\Omega }\left( 2\zeta _u^2+\frac{1}{2}\zeta _p^2+\zeta _q^2+\zeta _r^2+2(\zeta _u)_t^2\right) dxdt. \end{aligned}$$

(3.67)

Here we used the approximation property (2.14) of $\Pi _h$ as well as the optimal error of the initialization in Lemma 3.7, and the constant C depends on k, the final time T, $\Vert u\Vert _{L^\infty ((0,T);H^{k+4}(\Omega ))}$, $\Vert u_t\Vert _{{L^\infty ((0,T);H^{k+4}(\Omega ))}}$, and $\Vert u_{tt}\Vert _{L^\infty ((0,T);H^{k+2}(\Omega ))}$.

Finally we apply the Gronwall’s inequality and the triangle inequality, and reach the following theorem.

Theorem 3.8

For the semi-discrete DG scheme (3.24)–(3.27) with the numerical fluxes (3.28), where the parameters satisfy $\alpha ^2+\beta _1\beta _2=\frac{1}{4}$ and $\beta _i\ge 0$, $i=1, 2$, the following error estimates hold when the exact solution u of the Eq. (1.2) is sufficiently smooth,

$$\begin{aligned}&\Vert u-u_h\Vert ^2_\Omega +\Vert p-p_h\Vert ^2_\Omega +\Vert q-q_h\Vert ^2_\Omega +\Vert r-r_h\Vert ^2_\Omega +\Vert u_t-(u_h)_t\Vert ^2_\Omega \le Ch^{2k+2}. \end{aligned}$$

(3.68)

Here $r=u_x$, $q=r_x$, and $p=q_x$. And the constant C depends on k, the final time T, $\Vert u\Vert _{L^\infty ((0,T);H^{k+4}(\Omega ))}$, $\Vert u_t\Vert _{{L^\infty ((0,T);H^{k+4}(\Omega ))}}$, and $\Vert u_{tt}\Vert _{L^\infty ((0,T);H^{k+2}(\Omega ))}$.

3.3 Extension to General Even-Order Equations

The DG methods with the special family of numerical fluxes in the previous Sect. 3.2, as well as the theoretical analysis for stability and optimal error estimates can be extended to the general even-order PDEs in (1.3). The key lies in a careful choice of numerical fluxes. In this subsection, we will particularly give the formulation of the methods as well as the theoretical results for the sixth order equation (1.3) with $n=6$. Consider

$$\begin{aligned} u_t-u_x^{(6)}=0, \end{aligned}$$

(3.69)

with the periodic boundary condition. We first rewrite (3.69) into a first order system

$$\begin{aligned} u_t-u_x^5=0,\;u^5=u_x^4,\;u^4=u^3_x,\;u^3=u_x^2,\;u^2=u_x^1,\;u^1=u_x, \end{aligned}$$

(3.70)

then apply a DG method: look for $(u_h,u_h^{5},u_h^4,u_h^3,u_h^2,u_h^1)$ with $u_h\in {\mathcal {V}}_h^k,u_h^i \in {\mathcal {V}}_h^k,i=1,\ldots ,5$ such that for any $(v,v^{5},v^{4},v^3,v^{2},v^1)$ with $v\in {\mathcal {V}}_h^k,v^i\in {\mathcal {V}}_h^k,i=1,\ldots ,5$ and j

$$\begin{aligned} ((u_h)_t,v)_{I_j}+(u_h^{5},v_x)_{I_j}-((\widehat{u_h^{5}}v^-)_{j+\frac{1}{2}}+(\widehat{u_h^{5}}v^+)_{j-\frac{1}{2}}&=0,\end{aligned}$$

(3.71)

$$\begin{aligned} (u_h^{5},v^{5})_{I_j}+(u_h^{4},v^{5}_x)_{I_j}-(\widehat{u_h^{4}}(v^{5})^-)_{j+\frac{1}{2}}+(\widehat{u_h^{4}}(v^{5})^+)_{j-\frac{1}{2}}&=0,\end{aligned}$$

(3.72)

$$\begin{aligned} (u_h^{4},v^{4})_{I_j}+(u_h^{3},v^{4}_x)_{I_j}-(\widehat{u_h^{3}}(v^{4})^-)_{j+\frac{1}{2}}+(\widehat{u_h^{3}}(v^{4})^+)_{j-\frac{1}{2}}&=0,\end{aligned}$$

(3.73)

$$\begin{aligned} (u_h^{3},v^{3})_{I_j}+(u_h^{2},v^{3}_x)_{I_j}-(\widehat{u_h^{2}}(v^{3})^-)_{j+\frac{1}{2}}+(\widehat{u_h^{2}}(v^{3})^+)_{j-\frac{1}{2}}&=0,\end{aligned}$$

(3.74)

$$\begin{aligned} (u_h^{2},v^{2})_{I_j}+(u_h^{1},v^{2}_x)_{I_j}-(\widehat{u_h^{1}}(v^{2})^-)_{j+\frac{1}{2}}+(\widehat{u_h^{1}}(v^{2})^+)_{j-\frac{1}{2}}&=0,\end{aligned}$$

(3.75)

$$\begin{aligned} (u_h^{1},v^{1})_{I_j}+(u_h,v^{1}_x)_{I_j}-(\widehat{u_h}(v^{1})^-)_{j+\frac{1}{2}}+(\widehat{u_h}(v^{1})^+)_{j-\frac{1}{2}}&=0. \end{aligned}$$

(3.76)

Here, $\widehat{u_h}$ and $\widehat{u_h^{i}},i=1,\ldots ,5$ are numerical fluxes defined as

$$\begin{aligned} \widehat{u_h^{5}}&=F_1(u_h^{5},u_h,\beta _1),\;\widehat{u_h^{4}}=F_2(u_h^{4},u_h^{1},-\beta _2),\;\widehat{u_h^{3}}=F_1(u_h^{3},u_h^{2},\beta _1),\nonumber \\ \widehat{u_h^{2}}&=F_2(u_h^{2},u_h^{3},\beta _2),\;\widehat{u_h^{1}}=F_1(u_h^{1},u_h^4,-\beta _1),\;\widehat{u_h}=F_2(u_h,u_h^{5},\beta _2). \end{aligned}$$

(3.77)

Under the similar assumptions for the parameters $\alpha , \beta _1, \beta _2$ in the numerical fluxes as for the fourth order problem in Sect. 3.2, and following the similar analysis and definitions of initial conditions, one can carry out the $L^2$ stability and optimal error estimates of the DG methods above. The next two theorems summarize the results without the proofs.

Theorem 3.9

With $\beta _1\ge 0,\beta _2\ge 0$, the semi-discrete DG scheme (3.71)–(3.76) with the numerical fluxes (3.77) for the sixth order equation (1.3) satisfies

$$\begin{aligned} \Vert {\varvec{{\mathcal {E}}_h}}(T)\Vert ^2\le \Vert {\varvec{{\mathcal {E}}_h}}(0)\Vert ^2, \end{aligned}$$

where $\Vert {\varvec{{\mathcal {E}}_h}}(T)\Vert ^2=\left( 8\Vert u_h\Vert ^2_\Omega +\Vert u_h^1\Vert ^2_\Omega +\Vert u_h^2\Vert ^2_\Omega +\Vert u_h^3\Vert ^2_\Omega +\Vert u_h^4\Vert ^2_\Omega +\Vert u_h^5\Vert ^2_\Omega +8\Vert (u_h)_t\Vert ^2_\Omega \right) |_{t=T}$.

Theorem 3.10

For the semi-discrete DG scheme (3.71)–(3.76) with the numerical fluxes (3.77), where the parameters satisfy $\alpha ^2+\beta _1\beta _2=\frac{1}{4}$ and $\beta _i\ge 0$, $i=1, 2$, the following error estimates hold when the exact solution u of the Eq. (1.3) with $n=6$ is sufficiently smooth,

$$\begin{aligned}&\Vert u-u_h\Vert ^2_\Omega +\Vert u^1-u_h^1\Vert ^2_\Omega +\Vert u^2-u_h^2\Vert ^2_\Omega +\Vert u^3-u_h^3\Vert ^2_\Omega +\Vert u^4-u_h^4\Vert ^2_\Omega \nonumber \\&\quad +\Vert u^5-u_h^5\Vert ^2_\Omega +\Vert u_t-(u_h)_t\Vert ^2_\Omega \le Ch^{2k+2}. \end{aligned}$$

(3.78)

Here $u^{i}=u^{(i-1)}_x, i=1, 2, \ldots , 5$. And the constant C depends on k, the final time T, $\Vert u\Vert _{L^\infty ((0,T);H^{k+6}(\Omega ))}$, $\Vert u_t\Vert _{{L^\infty ((0,T);H^{k+6}(\Omega ))}}$, $\Vert u_{tt}\Vert _{L^\infty ((0,T);H^{k+6}(\Omega ))}$.

4 DG Methods for the Third Order Wave Equation

In this section, we will propose a family of DG methods for the third order wave equation (1.4), with the optimally accurate LDG method in [30] as a special case. Following the analysis in [30], we will establish the $L^2$ stability and optimal error estimates of a sub-family of the proposed methods when the boundary conditions are periodic.

To formulate the DG methods for (1.4), we start with the first order form of the equation,

$$\begin{aligned} u_t + p_x=0, \quad p - q_x=0,\quad q -u_x=0. \end{aligned}$$

(4.1)

Based on this system, a DG method is to find $u_h$, $p_h$, $q_h$$\in {\mathcal {V}}_h^k$ such that for any $v$, $w$, $z$$\in {\mathcal {V}}_h^k$ and for any j,

$$\begin{aligned} \left( (u_h)_t,v\right) {_{I_j}}- (p_h,v_x){_{I_j}}+ \left( F_p v^-\right) _{{j+\frac{1}{2}}}-\left( F_pv^+\right) _{{j-\frac{1}{2}}}&=0, \end{aligned}$$

(4.2)

$$\begin{aligned} \left( p_h,w\right) {_{I_j}}+ (q_h,w_x){_{I_j}}- \left( F_q w^-\right) _{{j+\frac{1}{2}}} +\left( F_qw^+\right) _{{j-\frac{1}{2}}}&=0, \end{aligned}$$

(4.3)

$$\begin{aligned} \left( q_h,z\right) {_{I_j}}+ (u_h,z_x){_{I_j}}- \left( F_u z^-\right) _{{j+\frac{1}{2}}} +\left( F_uz^+\right) _{{j-\frac{1}{2}}}&=0. \end{aligned}$$

(4.4)

Here, $F_p$, $F_q$ and $F_u$ in (4.2)–(4.4) are single-valued numerical fluxes and they could take very general forms. To avoid overwhelmingly too many parameters, in this paper, we choose

$$\begin{aligned} F_p=F_1(p_h,u_h,\beta _1),\quad F_q=q_h^+,\quad F_u=F_2(u_h,p_h,\beta _2), \end{aligned}$$

(4.5)

that involve three parameters $\alpha , \beta _1,\beta _2$. The conditions on these parameters will be further specified along with our analysis for stability and error estimates.

By summing up (4.2)–(4.4) over j, we reach a compact form of the scheme: look for $u_h,p_h,q_h\in {\mathcal {V}}_h^k$ such that

$$\begin{aligned} B(u_h,p_h,q_h;v,w,z)=0, \quad \forall v,w,z\in {\mathcal {V}}_h^k, \end{aligned}$$

(4.6)

where

$$\begin{aligned} B(u_h,p_h,q_h;v,w,z)&= \int _{\Omega }(u_h)_tvdx- \sum \limits _{j}\left( \int _{I_j}p_hv_xdx + (F_p[v])_{j-\frac{1}{2}}\right) \nonumber \\&\quad +\int _{\Omega }p_hwdx+\sum \limits _{j}\left( \int _{I_j}q_hw_xdx + (F_q[w])_{j-\frac{1}{2}}\right) \nonumber \\&\quad +\int _{\Omega }q_hzdx+\sum \limits _{j}\left( \int _{I_j}u_hz_xdx + (F_u[z])_{j-\frac{1}{2}}\right) . \end{aligned}$$

(4.7)

4.1 $L^2$ Stability

In this subsection, we present the $L^2$ stability analysis for the DG scheme (4.6)–(4.7) with the numerical fluxes (4.5) under some assumptions on the parameters $\alpha , \beta _1$ and $\beta _2$. Similar to the analysis for the LDG method in [30], we first obtain four energy equations, and then prove the $L^2$ stability for the numerical solution $u_h$ and the auxiliary variables $p_h$, $q_h$. Note that just as for the fourth and sixth order equations, more than one energy equation is needed in order for us to later establish optimal error estimates.

$\bullet$The first energy equation To obtain the energy equation related to $\Vert u_h\Vert _\Omega$, we take the test functions $v=u_h$, $w=q_h$ and $z=-p_h$ in (4.6). Then, the following equality is obtained

$$\begin{aligned} 0=B(u_h,p_h,q_h;u_h,q_h,-p_h)&=\int _{\Omega }(u_h)_tu_hdx - \sum \limits _{j}\left( \int _{I_j}p_h(u_h)_xdx+(F_p[u_h])_{j-\frac{1}{2}}\right) \nonumber \\&\quad +\int _{\Omega }p_hq_hdx + \sum \limits _{j}\left( \int _{I_j}q_h(q_h)_xdx+(F_q[q_h])_{j-\frac{1}{2}}\right) \nonumber \\&\quad - \int _{\Omega }q_hp_hdx - \sum \limits _{j}\left( \int _{I_j}u_h(p_h)_xdx+(F_u[p_h])_{j-\frac{1}{2}}\right) \nonumber \\&=\frac{1}{2}\frac{d}{dt}\int _{\Omega }u_h^2dx +\sum \limits _{j}\left( \frac{1}{2}[q_h]^2_{j-\frac{1}{2}}-H_j(u_h,p_h)\right) . \end{aligned}$$

(4.8)

Here, we use the definition of $H_j$ in (3.8). Combing (4.8) with (3.9), we have

$$\begin{aligned}&B(u_h,p_h,q_h;u_h,q_h,-p_h)\nonumber \\&\quad =\frac{1}{2}\frac{d}{dt}\int _{\Omega }u_h^2dx+\sum \limits _{j}\left( \frac{1}{2}[q_h]^2-\beta _1[u_h]^2-\beta _2[p_h]^2\right) _{j-\frac{1}{2}}=0. \end{aligned}$$

(4.9)

$\bullet$The second energy equation In the next step, we take the time derivative in (4.3)–(4.4), sum them up with (4.2) over j and have

$$\begin{aligned} B(u_h,(p_h)_t,(q_h)_t;v,w,z)=0, \quad \forall v,w,z\in {\mathcal {V}}_h^k. \end{aligned}$$

(4.10)

By taking the test functions $v=-(q_h)_t$, $w=p_h$ and $z=(u_h)_t$ in (4.10), we obtain

$$\begin{aligned} 0&=B(u_h,(p_h)_t,(q_h)_t;-(q_h)_t,p_h,(u_h)_t)\nonumber \\&=-\int _{\Omega }(u_h)_t(q_h)_t dx + \sum \limits _{j}\left( \int _{I_j}p_h (q_h)_{tx}dx+(F_p[(q_h)_t])_{j-\frac{1}{2}}\right) \nonumber \\&\quad +\int _{\Omega }(p_h)_t p_h dx + \sum \limits _{j}\left( \int _{I_j}(q_h)_t(p_h)_xdx+((F_q)_t[p_h])_{j-\frac{1}{2}}\right) \nonumber \\&\quad + \int _{\Omega }(q_h)_t(u_h)_t dx + \sum \limits _{j}\left( \int _{I_j}(u_h)_t (u_h)_{tx}dx+((F_u)_t[(u_h)_t])_{j-\frac{1}{2}}\right) . \end{aligned}$$

(4.11)

Using the definition of $F_p$, $F_q$ and $F_u$ in (4.5), we have the second energy equation as

$$\begin{aligned} 0&=B(u_h,(p_h)_t,(q_h)_t;-(q_h)_t,p_h,(u_h)_t)=\frac{1}{2}\frac{d}{dt}\int _{\Omega }p_h^2dx \nonumber \\&\quad +\sum \limits _{j}\left( -\alpha [(u_h)_t]^2+\left( \alpha +\frac{1}{2}\right) [p_h][(q_h)_t]+\beta _1[u_h][(q_h)_t]+\beta _2[(p_h)_t][(u_h)_t]\right) _{j-\frac{1}{2}}. \end{aligned}$$

(4.12)

$\bullet$The third energy equation Taking the time derivative in (4.2)–(4.4) and summing them up over j, we get

$$\begin{aligned} B((u_h)_t,(p_h)_t,(q_h)_t;v,w,z)=0, \quad \forall v,w,z\in {\mathcal {V}}_h^k. \end{aligned}$$

(4.13)

Similar to Eq. (4.8), we take $v=(u_h)_t$, $w=(q_h)_t$ and $z=-(p_h)_t$ in (4.13). Using the definition of $F_p$, $F_q$ and $F_u$ in (4.5) and $H_j$ in (3.8), we obtain

$$\begin{aligned} 0&=B((u_h)_t,(p_h)_t,(q_h)_t;(u_h)_t,(q_h)_t,-(p_h)_t)\nonumber \\&=\frac{1}{2}\frac{d}{dt}\int _{\Omega }(u_h)_t^2dx +\sum \limits _{j}\left( \frac{1}{2}[(q_h)_t^2]_{j-\frac{1}{2}}- H_j((u_h)_t,(p_h)_t) \right) \nonumber \\&=\frac{1}{2}\frac{d}{dt}\int _{\Omega }(u_h)^2_tdx+\sum \limits _{j}\left( \frac{1}{2}[(q_h)_t]^2-\beta _1[(u_h)_t]^2-\beta _2[(p_h)_t]^2\right) _{j-\frac{1}{2}}. \end{aligned}$$

(4.14)

$\bullet$The fourth energy equation In this step, we take the time derivative in (4.4), sum it up with (4.2)–(4.3) over j, and have

$$\begin{aligned} B(u_h,p_h,(q_h)_t;v,w,z)=0, \quad \forall v,w,z\in {\mathcal {V}}_h^k. \end{aligned}$$

(4.15)

By taking the test function $v=0$, $w=\frac{1}{2}(u_h)_t$ and $z=\frac{1}{2}q_h$ in (4.15), we obtain

$$\begin{aligned} 0&=B\left( u_h,p_h,(q_h)_t;0,\frac{1}{2}(u_h)_t,\frac{1}{2}q_h\right) \nonumber \\&=\frac{1}{2}\int _{\Omega }p_h (u_h)_t dx + \frac{1}{2}\sum \limits _{j}\left( \int _{I_j}q_h(u_h)_{tx}dx+(F_q[(u_h)_t])_{j-\frac{1}{2}}\right) \nonumber \\&\quad + \frac{1}{2}\int _{\Omega }(q_h)_tq_h dx + \frac{1}{2}\sum \limits _{j}\left( \int _{I_j}(u_h)_t (q_h)_xdx+((F_u)_t[q_h])_{j-\frac{1}{2}}\right) . \end{aligned}$$

(4.16)

Using the definitions of $F_p$, $F_q$ and $F_u$ in (4.5), we finally have

$$\begin{aligned} 0=B(u_h,p_h,(q_h)_t;0,\frac{1}{2}(u_h)_t,\frac{1}{2}q_h)&=\frac{1}{2}\int _{\Omega }(p_h (u_h)_t+(q_h)_tq_h) dx \nonumber \\&\quad +\frac{1}{2}\sum \limits _{j}\left( (\frac{1}{2}{-}\alpha )[q_h][(u_h)_t]{+}\beta _2[(p_h)_t][q_h]\right) _{j-\frac{1}{2}}. \end{aligned}$$

(4.17)

Now we are ready to state the $L^2$ stability of the proposed DG methods.

Theorem 4.1

Under the conditions

$$\begin{aligned}&\alpha ^2+\beta _1\beta _2=\frac{1}{4},\quad -2\le \beta _1\le 0, \quad -8\le \beta _2\le 0,\nonumber \\&4\beta _1+4\alpha \le \beta _2,\quad \alpha +2\beta _2\le -\frac{1}{2},\quad 8\beta _1+7\alpha \le -\frac{1}{2}+\frac{3}{2}\beta _2, \end{aligned}$$

(4.18)

the semi-discrete DG method (4.6)–(4.7) with the numerical fluxes (4.5) satisfies

$$\begin{aligned}&\Vert u_h(T)\Vert ^2_\Omega +\Vert p_h(T)\Vert ^2_\Omega +\Vert (u_h)_t(T)\Vert ^2_\Omega +\frac{1}{2}\Vert q_h(T)\Vert ^2_\Omega \nonumber \\&\quad \le e^{\frac{T}{2}}\left( \Vert u_h(0)\Vert ^2_\Omega +\Vert p_h(0)\Vert ^2_\Omega +\Vert (u_h)_t(0)\Vert ^2_\Omega +\frac{1}{2}\Vert q_h(0)\Vert ^2_\Omega \right) , \end{aligned}$$

(4.19)

where T is the final time. Particularly, when $\alpha =-\frac{1}{2}$ and $\beta _1=\beta _2=0$, the stability result (4.19) can be replaced by

$$\begin{aligned}&\Vert u_h(T)\Vert ^2_\Omega +\Vert p_h(T)\Vert ^2_\Omega +\Vert (u_h)_t(T)\Vert ^2_\Omega +\frac{1}{2}\Vert q_h(T)\Vert ^2_\Omega \nonumber \\&\quad \le \Vert u_h(0)\Vert ^2_\Omega +\left( \frac{T}{2}+1\right) \Vert p_h(0)\Vert ^2_\Omega +\left( \frac{T}{2}+1\right) \Vert (u_h)_t(0)\Vert ^2_\Omega +\frac{1}{2}\Vert q_h(0)\Vert ^2_\Omega . \end{aligned}$$

(4.20)

Proof

Summing up the four energy equations (4.9), (4.12), (4.14) and (4.17), we have

$$\begin{aligned} 0&=\frac{1}{2}\frac{d}{dt}\left( \Vert u_h\Vert ^2_\Omega +\Vert p_h\Vert ^2_\Omega +\Vert (u_h)_t\Vert ^2_\Omega +\frac{1}{2}\Vert q_h\Vert ^2_\Omega \right) +\frac{1}{2}(p_h,(u_h)_t)\nonumber \\&\quad +\sum \limits _{j}\left( \frac{1}{2}[q_h]^2{+}(-\alpha {-}\beta _1)[(u_h)_t]^2{+}\frac{1}{2}[(q_h)_t]^2{-}\beta _1[u_h]^2{-}\beta _2[p_h]^2{-}\beta _2[(p_h)_t]^2\right) _{j-\frac{1}{2}}\nonumber \\&\quad +\sum \limits _{j}\left( (\alpha +\frac{1}{2})[p_h][(q_h)_t]+\beta _2[(p_h)_t][(u_h)_t]+\beta _1[u_h][(q_h)_t]\right) _{j-\frac{1}{2}}\nonumber \\&\quad +\frac{1}{2}\sum \limits _{j}\left( (\frac{1}{2}-\alpha )[q_h][(u_h)_t]+\beta _2[(p_h)_t][q_h]\right) _{j-\frac{1}{2}}. \end{aligned}$$

(4.21)

Now we introduce two symmetric matrices $S_1$ and $S_2$

$$\begin{aligned} S_1= \begin{pmatrix} - \beta _1 &{} 0 &{}\frac{\beta _1}{2} \\ 0 &{} -\beta _2 &{}\frac{\alpha +\frac{1}{2}}{2}\\ \frac{\beta _1}{2}&{}\frac{\alpha +\frac{1}{2}}{2}&{}\frac{1}{2} \end{pmatrix} ,\quad S_2= \begin{pmatrix} &{}\frac{1}{2} &{} \frac{\frac{1}{2}-\alpha }{4} &{}\frac{\beta _2}{4}\\ &{} \frac{\frac{1}{2}-\alpha }{4} &{}-\alpha -\beta _1 &{} \frac{\beta _2}{2} \\ &{}\frac{\beta _2}{4}&{}\frac{\beta _2}{2}&{}-\beta _2\\ \end{pmatrix} \end{aligned}$$

(4.22)

and a set of vector-valued functions ${\mathbf {U}}_{1j}$ and ${\mathbf {U}}_{2j}$, $j=1,\ldots ,N$,

$$\begin{aligned} {\mathbf {U}}_{1j}^\text {T}=([u_h],[p_h],[(q_h)_t])_{j-\frac{1}{2}}, \qquad {\mathbf {U}}_{2j}^\text {T}=([q_h],[(u_h)_t],[(p_h)_t])_{j-\frac{1}{2}}. \end{aligned}$$

(4.23)

Then, the Eq. (4.21) can be rewritten into

$$\begin{aligned} 0&=\frac{1}{2}\frac{d}{dt}\left( \Vert u_h\Vert ^2_\Omega +\Vert p_h\Vert ^2_\Omega +\Vert (u_h)_t\Vert ^2_\Omega +\frac{1}{2}\Vert q_h\Vert ^2_\Omega \right) +\frac{1}{2}(p_h,(u_h)_t)\nonumber \\&\quad +\sum \limits _{j}\left( {\mathbf {U}}_{1j}^\text {T}S_1{\mathbf {U}}_{1j}+{\mathbf {U}}_{2j}^\text {T}S_2 {\mathbf {U}}_{2j} \right) . \end{aligned}$$

(4.24)

In order to obtain conditions on $\alpha ,\beta _1$ and $\beta _2$ such that both $S_1$ and $S_2$ are positive semi-definite, we follow the sufficient and necessary condition “all the k-th principal minors are nonnegative, with $k=1, 2, 3$”, as well as the relation $\alpha ^2+\beta _1\beta _2=\frac{1}{4}$ (to simplify the conditions), the details are given in “Appendix B”. This leads to (4.18), and under these conditions (4.24) becomes

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt}\left( \Vert u_h\Vert ^2_\Omega +\Vert p_h\Vert ^2_\Omega +\Vert (u_h)_t\Vert ^2_\Omega +\frac{1}{2}\Vert q_h\Vert ^2_\Omega \right) +\frac{1}{2}(p_h,(u_h)_t)\le 0. \end{aligned}$$

(4.25)

Now apply $|(p_h,(u_h)_t)|\le \frac{1}{2} (\Vert p_h\Vert ^2_\Omega +\Vert (u_h)_t\Vert ^2_\Omega )$, and we can obtain

$$\begin{aligned}&\frac{d}{dt}\left( \Vert u_h\Vert ^2_\Omega +\Vert p_h\Vert ^2_\Omega +\Vert (u_h)_t\Vert ^2_\Omega +\frac{1}{2}\Vert q_h\Vert ^2_\Omega \right) \le \frac{1}{2}\left( \Vert p_h\Vert ^2_\Omega +\Vert (u_h)_t\Vert ^2_\Omega \right) \end{aligned}$$

(4.26)

$$\begin{aligned}&\quad \le \frac{1}{2}\left( \Vert u_h\Vert ^2_\Omega +\Vert p_h\Vert ^2_\Omega +\Vert (u_h)_t\Vert ^2_\Omega +\frac{1}{2}\Vert q_h\Vert ^2_\Omega \right) . \end{aligned}$$

(4.27)

The stability result in (4.19) follows from the Gronwall’s inequality.

Finally we consider a special case when $\alpha =-\frac{1}{2}$ and $\beta _1=\beta _2=0$. Note that for this case, (4.12) and (4.14) imply

$$\begin{aligned} \Vert p_h(t)\Vert _\Omega \le \Vert p_h(0)\Vert _\Omega , \qquad \Vert (u_h)_t(t)\Vert _\Omega \le \Vert (u_h)_t(0)\Vert _\Omega , \;\;\forall t>0. \end{aligned}$$

(4.28)

Integrating (4.26) over [0, T], we get

$$\begin{aligned}&\Vert u_h(T)\Vert ^2_\Omega +\Vert p_h(T)\Vert ^2_\Omega +\Vert (u_h)_t(T)\Vert ^2_\Omega +\frac{1}{2}\Vert q_h(T)\Vert ^2_\Omega \nonumber \\&\quad \le \frac{1}{2}\int _0^T\left( \Vert p_h\Vert ^2_\Omega (t)+\Vert (u_h)_t(t)\Vert ^2_\Omega \right) dt+\Vert u_h(0)\Vert ^2_\Omega +\Vert p_h(0)\Vert ^2_\Omega +\Vert (u_h)_t(0)\Vert ^2_\Omega +\frac{1}{2}\Vert q_h(0)\Vert ^2_\Omega . \end{aligned}$$

(4.29)

The stability relation (4.20) follows from (4.29) and (4.28). $\square$

Remark 4.2

When $\alpha =-\frac{1}{2}$ and $\beta _1=\beta _2=0$, our proposed DG method will become the LDG method in [30] with one set of alternating numerical fluxes. And the stability result (4.20) was also established in [30].

We want to point out that the parameter conditions (4.18) do not include another set of alternating fluxes, that is, the numerical fluxes (4.5) with $\alpha =\frac{1}{2},\beta _1=\beta _2=0$, or equivalently, $F_p=p_h^+,F_q=q_h^+, F_u=u_h^-$. The corresponding DG method has quite different properties from that in [30] and also from those in Theorem 4.1, and its $L^2$ stability needs to be established separately. In the next Theorem, we state the energy stability result for this somewhat different DG method, and the proof is given in “Appendix A”.

Theorem 4.3

Use the numerical fluxes (4.5) with $\alpha =\frac{1}{2},\beta _1=\beta _2=0$, the semi-discrete DG scheme (4.6)–(4.7) satisfies the energy stability

$$\begin{aligned}&\Vert u_h(T)\Vert ^2_\Omega +\frac{1}{2}\Vert p_h(T)\Vert ^2_\Omega +\Vert (u_h)_t(T)\Vert ^2_\Omega +\Vert q_h(T)\Vert ^2_\Omega +\Vert (q_h)_t(T)\Vert ^2_\Omega \nonumber \\&\quad \le \Vert u_h(0)\Vert ^2_\Omega +\frac{1}{2}\Vert p_h(0)\Vert ^2_\Omega +\left( \frac{T}{2}+1\right) \Vert (u_h)_t(0)\Vert ^2_\Omega +\Vert q_h(0)\Vert ^2_\Omega +\left( \frac{T}{2}+1\right) \Vert (q_h)_t(0)\Vert ^2_\Omega . \end{aligned}$$

(4.30)

4.2 $L^2$ Error Estimates

In this subsection, the optimal a priori$L^2$ error estimates will be proved for the DG method (4.6)–(4.7) for the third order equation when the exact solution is sufficiently smooth, under the conditions in (4.18) on the parameters $\alpha , \beta _1, \beta _2$ in the numerical fluxes (4.5). Particularly, the relation $\alpha ^2+\beta _1\beta _2=\frac{1}{4}$ holds. Since the numerical fluxes (4.5) are consistent, the exact solution u, $q=u_x$, and $p=q_x$ satisfy

$$\begin{aligned} B(u,p,q;v,w,z)=0, \quad \forall v,w,z\in {\mathcal {V}}_h^k, \end{aligned}$$

(4.31)

therefore we get the error equation

$$\begin{aligned} B(e_u,e_p,e_q;v,w,z)=0,\; \forall v,w,z\in {\mathcal {V}}_h^k. \end{aligned}$$

(4.32)

Here $e_\phi =\phi -\phi _h$, with $\phi =u, p, q$, are the error functions. In order to obtain the optimal error estimates, the following projection is used

$$\begin{aligned} \Pi _h \begin{pmatrix} p\\ u\\ \end{pmatrix} = \begin{pmatrix} \Pi _h^1 (p,u,\beta _1)\\ \Pi _h^2 (u,p,\beta _2)\\ \end{pmatrix}, \end{aligned}$$

(4.33)

and $e_u, e_p$ will be decomposed into $e_\phi =\eta _\phi +\zeta _\phi , \phi =u, p$ based on (2.10)–(2.11), while $e_q=\eta _q+\zeta _q$ where $\eta _q=q-P_h^+q$ and $\zeta _q=P_h^+q-q_h$.

For the third order equation (1.4), we choose the initial condition $p_h(x,0)=P_h^- p(x,0)$ with $p(x,0)=u_{xx}(x,0)$. Based on $p_h(x,0)$, we can further define the initial data $q_h(x,0), u_h(x, 0) \in {\mathcal {V}}_h^k$, satisfying

$$\begin{aligned}&(p_h,w)_{I_j}+(q_h,w_x)|_{I_j}-({\widehat{q}}_hw^-)|_{j+\frac{1}{2}}+({\widehat{q}}_hw^+)|_{j-\frac{1}{2}}=0,\; \forall w\in {\mathcal {V}}_h^k,\; \nonumber \\&\quad \text{ with }\;{\widehat{q}}_h=q_h^+(x,0) \;\;\text{ and }\;\; \int _\Omega q_h(x,0)dx=\int _\Omega q(x,0)dx, \end{aligned}$$

(4.34)

$$\begin{aligned}&(q_h,w)_{I_j}+(u_h,w_x)|_{I_j}-({\widehat{u}}_hw^-)|_{j+\frac{1}{2}}+({\widehat{u}}_hw^+)|_{j-\frac{1}{2}}=0,\; \forall w\in {\mathcal {V}}_h^k,\; \nonumber \\&\quad \text{ with }\;{\widehat{u}}_h=u_h^+(x,0) \;\;\text{ and }\;\; \int _\Omega u_h(x,0)dx=\int _\Omega u(x,0)dx. \end{aligned}$$

(4.35)

Similar to the analysis for Lemma 5.1 in [23] and Lemma 2.4 in [30], the following lemma can be established.

Lemma 4.4

Assuming u(x, 0) is sufficiently smooth, the initial conditions described above are well defined and satisfy the following estimates

$$\begin{aligned}&||p(x,0)-p_h(x,0)||_\Omega \le Ch^{k+1},\; ||q(x,0)-q_h(x,0)||_\Omega \le Ch^{k+1},\nonumber \\&||u(x,0)-u_h(x,0)||_\Omega \le Ch^{k+1},\; ||u_t(x,0)-(u_h)_t(x,0)||_\Omega \le Ch^{k+1}, \end{aligned}$$

(4.36)

Here $(u_h)_t(x,0)$ is determined by (4.2) with $F_p=p_h^-(x,0)$ at $t=0$. And C depends on $||u(x,0)||_{H^{k+3}(\Omega )}$ and $q(x,0)=u_x(x,0)$.

To obtain the optimal error estimates, we follow the idea of the energy stability analysis and get four important error equations.

4.2.1 The First Error Equation

Since B is linear, the error equation (4.32) can be written as

$$\begin{aligned} B(\zeta _u,\zeta _p,\zeta _q;v,w,z)=-B(\eta _u,\eta _p,\eta _q;v,w,z), \quad \forall v,w,z\in {\mathcal {V}}_h^k. \end{aligned}$$

(4.37)

We then take the test functions $v=\zeta _u$, $w=\zeta _q$ and $z=-\zeta _p$, all from ${\mathcal {V}}_h^k$, and get

$$\begin{aligned} B(\zeta _u,\zeta _p, \zeta _q;\zeta _u,\zeta _q, -\zeta _p)&=\frac{1}{2}\frac{d}{dt}\int _{\Omega }\zeta _u^2dx+\sum \limits _{j}\left( \frac{1}{2}[\zeta _q]^2-\beta _1[\zeta _u]^2-\beta _2[\zeta _p]^2\right) _{j-\frac{1}{2}}, \end{aligned}$$

(4.38)

$$\begin{aligned} B(\eta _u,\eta _p, \eta _q;\zeta _u,\zeta _q, -\zeta _p)&=\int _{\Omega }(\eta _u)_t\zeta _udx - \sum \limits _{j}\left( \int _{I_j}\eta _p(\zeta _u)_xdx+(F_1(\eta _p,\eta _u, \beta _1)[\zeta _u])_{j-\frac{1}{2}}\right) \nonumber \\&\quad +\int _{\Omega }\eta _p\zeta _qdx + \sum \limits _{j}\left( \int _{I_j}\eta _q(\zeta _q)_xdx+((\eta _q)^+[\zeta _q])_{j-\frac{1}{2}}\right) \nonumber \\&\quad - \int _{\Omega }\eta _q\zeta _pdx - \sum \limits _{j}\left( \int _{I_j}\eta _u(\zeta _p)_xdx+(F_2(\eta _u,\eta _p, \beta _2)[\zeta _p])_{j-\frac{1}{2}}\right) \nonumber \\&=\int _{\Omega }\left( (\eta _u)_t\zeta _u+\eta _p\zeta _q-\eta _q\zeta _p\right) dx. \end{aligned}$$

(4.39)

Here we have used (2.6) and the properties (2.13) and (2.16) in Lemma 2.1. Now combining (4.37)–(4.39), we obtain

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt}\int _{\Omega }\zeta _u^2dx+\sum \limits _{j}\left( \frac{1}{2}[\zeta _q]^2-\beta _1[\zeta _u]^2-\beta _2[\zeta _p]^2\right) _{j-\frac{1}{2}}\nonumber \\&\quad =-\int _{\Omega }\left( (\eta _u)_t\zeta _u+\eta _p\zeta _q-\eta _q\zeta _p\right) dx. \end{aligned}$$

(4.40)

4.2.2 The Second Error Equation

Following the similar procedure to derive (4.10) in the stability analysis, we get an error equation in the following form,

$$\begin{aligned} B(\zeta _u,(\zeta _p)_t,(\zeta _q)_t;v,w,z)=-B(\eta _u,(\eta _p)_t,(\eta _q)_t;v,w,z), \quad \forall v,w,z\in {\mathcal {V}}_h^k. \end{aligned}$$

(4.41)

Now we take the test functions $v=-(\zeta _q)_t$, $w=\zeta _p$ and $z=(\zeta _u)_t$, use the property (2.6) of $P_h^+$ and (2.13), (2.16) in Lemma 2.1, and obtain

$$\begin{aligned}&B(\zeta _u,(\zeta _p)_t,(\zeta _q)_t;-(\zeta _q)_t,\zeta _p,(\zeta _u)_t)=\frac{1}{2}\frac{d}{dt}\int _{\Omega }\zeta _p^2dx \nonumber \\&\quad +\sum \limits _{j}\left( -\alpha [(\zeta _u)_t]^2+(\alpha +\frac{1}{2})[\zeta _p][(\zeta _q)_t]+\beta _1[\zeta _u][(\zeta _q)_t]+\beta _2[(\zeta _p)_t][(\zeta _u)_t]\right) _{j-\frac{1}{2}},\end{aligned}$$

(4.42)

$$\begin{aligned}&B(\eta _u,(\eta _p)_t, (\eta _q)_t;-(\zeta _q)_t,\zeta _p,(\zeta _u)_t)=\int _{\Omega }\left( -(\eta _u)_t(\zeta _q)_t+(\eta _p)_t\zeta _p+(\eta _q)_t(\zeta _u)_t\right) dx. \end{aligned}$$

(4.43)

They, combined with (4.41), will lead to

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt}\int _{\Omega }\zeta _p^2dx+\sum \limits _{j}\left( -\alpha [(\zeta _u)_t]^2+(\alpha +\frac{1}{2})[\zeta _p][(\zeta _q)_t]+\beta _1[\zeta _u][(\zeta _q)_t]+\beta _2[(\zeta _p)_t][(\zeta _u)_t]\right) _{j-\frac{1}{2}}\nonumber \\&\quad =-\int _{\Omega }\left( -(\eta _u)_t(\zeta _q)_t+(\eta _p)_t\zeta _p+(\eta _q)_t(\zeta _u)_t\right) dx. \end{aligned}$$

(4.44)

4.2.3 The Third Error Equation

Similar to how we derive (4.13), we can have the following error equation

$$\begin{aligned} B((\zeta _u)_t,(\zeta _p)_t,(\zeta _q)_t;v,w,z){=}-B((\eta _u)_t,(\eta _p)_t,(\eta _q)_t;v,w,z), \quad \forall v,w,z\in {\mathcal {V}}_h^k. \end{aligned}$$

(4.45)

With the test functions taken to be $v=(\zeta _u)_t$, $w=(\zeta _q)_t$ and $z=-(\zeta _p)_t$, the terms in (4.45) become

$$\begin{aligned}&B((\zeta _u)_t,(\zeta _p)_t,(\zeta _q)_t;(\zeta _u)_t,(\zeta _q)_t,-(\zeta _p)_t)\nonumber \\&\quad =\frac{1}{2}\frac{d}{dt}\int _{\Omega }(\zeta _u)^2_tdx+\sum \limits _{j}\left( \frac{1}{2}[(\zeta _q)_t]^2-\beta _1[(\zeta _u)_t]^2-\beta _2[(\zeta _p)_t]^2\right) _{j-\frac{1}{2}}, \end{aligned}$$

(4.46)

$$\begin{aligned}&B((\eta _u)_t,(\eta _p)_t,(\eta _q)_t;(\zeta _u)_t,(\zeta _q)_t,-(\zeta _p)_t)=\int _{\Omega }\left( (\eta _u)_{tt}(\zeta _u)_t+ (\eta _p)_t(\zeta _q)_t-(\eta _q)_t(\zeta _p)_t\right) dx. \end{aligned}$$

(4.47)

Combining (4.45) with (4.46)–(4.47), we obtain

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt}\int _{\Omega }(\zeta _u)_t^2dx+\sum \limits _{j}\left( \frac{1}{2}[(\zeta _q)_t]^2-\beta _1[(\zeta _u)_t]^2-\beta _2[(\zeta _p)_t]^2\right) _{j-\frac{1}{2}}\nonumber \\&\quad =-\int _{\Omega }\left( (\eta _u)_{tt}(\zeta _u)_t+(\eta _p)_t(\zeta _q)_t-(\eta _q)_t(\zeta _p)_t\right) dx. \end{aligned}$$

(4.48)

4.2.4 The Fourth Error Equation

Last, we use the error equation

$$\begin{aligned} B(\zeta _u,\zeta _p,(\zeta _q)_t;v,w,z)=-B(\eta _u,\eta _p,(\eta _q)_t;v,w,z), \quad \forall v,w,z\in {\mathcal {V}}_h^k. \end{aligned}$$

(4.49)

Similar to the Eq. (4.16), we take the test functions $v=0$, $w=\frac{1}{2}(\zeta _u)_t$ and $z=\frac{1}{2}\zeta _q$ in (4.49), and obtain

$$\begin{aligned} B\left( \zeta _u,\zeta _p,(\zeta _q)_t;0,\frac{1}{2}(\zeta _u)_t,\frac{1}{2}\zeta _q\right)&=\frac{1}{2}\int _{\Omega }\left( \zeta _p (\zeta _u)_t+(\zeta _q)_t\zeta _q\right) dx\nonumber \\&\quad +\frac{1}{2}\sum \limits _{j}\left( \left( \frac{1}{2}-\alpha \right) [\zeta _q][(\zeta _u)_t]+\beta _2[(\zeta _p)_t][\zeta _q]\right) _{j-\frac{1}{2}}, \end{aligned}$$

(4.50)

and

$$\begin{aligned} B\left( \eta _u,\eta _p,(\eta _q)_t;0,\frac{1}{2}(\zeta _u)_t,\frac{1}{2}\zeta _q\right)&=\frac{1}{2}\int _{\Omega }\left( \eta _p(\zeta _u)_t +(\eta _q)_t\zeta _q \right) dx, \end{aligned}$$

(4.51)

hence we have

$$\begin{aligned}&\frac{1}{2}\int _{\Omega }(\zeta _p (\zeta _u)_t+(\zeta _q)_t\zeta _q) dx+\frac{1}{2}\sum \limits _{j}\left( \left( \frac{1}{2}-\alpha \right) [\zeta _q][(\zeta _u)_t]+\beta _2[(\zeta _p)_t][\zeta _q]\right) _{j-\frac{1}{2}}\nonumber \\&\quad =-\frac{1}{2}\int _{\Omega }\left( \eta _p(\zeta _u)_t +(\eta _q)_t\zeta _q \right) dx. \end{aligned}$$

(4.52)

We are now ready to establish the optimal error estimates for the proposed DG methods.

Theorem 4.5

For the semi-discrete DG scheme (4.6)–(4.7) with the numerical fluxes (4.5), where the parameters satisfy (4.18), the following error estimates hold when the exact solution u of the Eq. (1.4) is sufficiently smooth,

$$\begin{aligned} \Vert e_u\Vert ^2_\Omega +\Vert e_p\Vert ^2_\Omega +\Vert e_q\Vert ^2_\Omega +\Vert (e_u)_t\Vert ^2_\Omega \le Ch^{2k+2}. \end{aligned}$$

(4.53)

Here $q=u_x, p=q_x$, and the constant C depends on k, the final time T, $\Vert u\Vert _{L^\infty ((0,T);{H^{k+3}(\Omega )})}$, $\Vert u_t\Vert _{L^\infty ((0,T);{H^{k+3}(\Omega )})}$ and $\Vert u_{tt}\Vert _{L^\infty ((0,T);{H^{k+3}(\Omega )})}$.

Proof

Summing up the four error equations (4.40), (4.44), (4.48) and (4.52), we have

$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\left( \Vert \zeta _u\Vert _\Omega ^2+\Vert \zeta _p\Vert _\Omega ^2+\Vert (\zeta _u)_t\Vert _\Omega ^2+\frac{1}{2}\Vert \zeta _q\Vert _\Omega ^2\right) +\sum \limits _{j}\left( \mathbf{V}_{1j}^\text {T}S_1 \mathbf{V}_{1j}+\mathbf{V}_{2j}^\text {T}S_2 \mathbf{V}_{2j}\right) ={\mathbf {F}}+{\mathbf {G}}, \end{aligned}$$

where $\mathbf{V}_{1j}^\text {T}=([\zeta _u],[\zeta _p],[(\zeta _q)_t])_{j-\frac{1}{2}}$, $\mathbf{V}_{2j}^\text {T}=([\zeta _q],[(\zeta _u)_t],[(\zeta _p)_t])_{j-\frac{1}{2}}$ with $j=1,\ldots ,N$ and

$$\begin{aligned} {\mathbf {F}}&=-\int _{\Omega }\Big ( (\eta _u)_t\zeta _u+\eta _p\zeta _q-\zeta _p\eta _q+(\eta _p)_t\zeta _p+(\eta _q)_t(\zeta _u)_t+(\eta _u)_{tt}(\zeta _u)_t\nonumber \\&\quad +\frac{1}{2}\eta _p(\zeta _u)_t +\frac{1}{2}(\eta _q)_t\zeta _q \Big )dx-\frac{1}{2}\int _{\Omega }\zeta _p(\zeta _u)_tdx, \end{aligned}$$

(4.54)

$$\begin{aligned} {\mathbf {G}}&=-\int _{\Omega }\left( (\eta _p)_t(\zeta _q)_t-(\zeta _p)_t(\eta _q)_t-(\eta _u)_t(\zeta _q)_t\right) dx. \end{aligned}$$

(4.55)

For ${\mathbf {F}}$, we can bound it by

$$\begin{aligned} |{\mathbf {F}}|\le C h^{2k+2}+\frac{1}{2}\left( \Vert \zeta _u\Vert ^2_\Omega +\frac{1}{4}\Vert \zeta _q\Vert ^2_\Omega +\frac{3}{4}\Vert \zeta _p\Vert ^2_\Omega +\Vert (\zeta _u)_t\Vert ^2_\Omega \right) . \end{aligned}$$

(4.56)

For ${\mathbf {G}}$, we integrate it in t over [0, T] , apply an integration by parts, and get

$$\begin{aligned} \int _0^T{\mathbf {G}}dt&=\int _{\Omega }\left( ((\eta _u)_t-(\eta _p)_t)\zeta _q+(\eta _q)_t\zeta _p\right) dx\mid _0^T -\int _0^T\int _{\Omega }\left( ((\eta _u)_{tt}-(\eta _p)_{tt})\zeta _q+(\eta _q)_{tt}\zeta _p\right) dxdt\nonumber \\&\le Ch^{2k+2}+\frac{1}{4}\left( \Vert \zeta _p\Vert ^2_\Omega +\frac{1}{2}\Vert \zeta _q\Vert ^2_\Omega \right) |_{t=T}+\frac{1}{8}\int ^T_0 \left( \Vert \zeta _p\Vert ^2_\Omega +\Vert \zeta _q\Vert ^2_\Omega \right) dt. \end{aligned}$$

(4.57)

Here, we have used the property of projections $\Pi _h$ and $P_h^+$, as well as the optimal initial error estimates in Lemma 4.4.

Combining the two estimates above, we have

$$\begin{aligned} \int _0^T( \mathbf {F+G})dt&\le C h^{2k+2}+\frac{1}{2}\int _0^T\left( \Vert \zeta _u\Vert ^2_\Omega +\frac{1}{2}\Vert \zeta _q\Vert ^2_\Omega +\Vert \zeta _p\Vert ^2_\Omega +\Vert (\zeta _u)_t\Vert ^2_\Omega \right) dt\nonumber \\&\quad +\left( \frac{1}{4}\Vert \zeta _p\Vert ^2_\Omega +\frac{1}{8}\Vert \zeta _q\Vert ^2_\Omega \right) |_{t=T}. \end{aligned}$$

(4.58)

Recall that $S_1$ and $S_2$ are positive semi-definite under the conditions (4.18), and this further gives

$$\begin{aligned}&\frac{1}{2}\left( \Vert \zeta _u\Vert ^2_\Omega +\frac{1}{2}\Vert \zeta _q\Vert ^2_\Omega +\Vert \zeta _p\Vert ^2_\Omega +\Vert (\zeta _u)_t\Vert ^2_\Omega \right) |_{t=T} \nonumber \\&\quad \le Ch^{2k+2} + \frac{1}{2}\int _0^T\left( \Vert \zeta _u\Vert ^2_\Omega +\frac{1}{2}\Vert \zeta _q\Vert ^2_\Omega +\Vert \zeta _p\Vert ^2_\Omega +\Vert (\zeta _u)_t\Vert ^2_\Omega \right) dt\nonumber \\&+\left( \frac{1}{4}\Vert \zeta _p\Vert ^2_\Omega +\frac{1}{8}\Vert \zeta _q\Vert ^2_\Omega \right) |_{t=T}, \end{aligned}$$

(4.59)

and therefore

$$\begin{aligned}&\frac{1}{8}\left( \Vert \zeta _u\Vert ^2_\Omega +\Vert \zeta _q\Vert ^2_\Omega +\Vert \zeta _p\Vert ^2_\Omega +\Vert (\zeta _u)_t\Vert ^2_\Omega \right) |_{t=T}\nonumber \\&\quad \le Ch^{2k+2} + \frac{1}{2}\int _0^T\left( \Vert \zeta _u\Vert ^2_\Omega +\Vert \zeta _q\Vert ^2_\Omega +\Vert \zeta _p\Vert ^2_\Omega +\Vert (\zeta _u)_t\Vert ^2_\Omega \right) dt. \end{aligned}$$

(4.60)

Now we can apply the Gronwall’s inequality, and reach

$$\begin{aligned}&\Vert \zeta _u\Vert ^2_\Omega +\Vert \zeta _q\Vert ^2_\Omega +\Vert \zeta _p\Vert ^2_\Omega +\Vert (\zeta _u)_t\Vert ^2_\Omega \le Ch^{2k+2}. \end{aligned}$$

(4.61)

Throughout the proof, the constant C depends on k, the final time T, $\Vert u\Vert _{L^\infty ((0,T);{H^{k+3}(\Omega )})}$, $\Vert u_t\Vert _{L^\infty ((0,T);{H^{k+3}(\Omega )})}$ and $\Vert u_{tt}\Vert _{L^\infty ((0,T);{H^{k+3}(\Omega )})}$. Finally, we can get the error estimates (4.53) by combining (4.61) with the projection errors (2.8) and (2.14). $\square$

Remark 4.6

We point out that for the semi-discrete DG method with the alternating flux $\alpha =\frac{1}{2},\beta _1=\beta _2=0$ (associated with Theorem 4.3), we just get its suboptimal error estimates because of the difficulty to find the well-defined initial conditions satisfying $||(q_t)(x,0)-(q_h)_t(x,0)||^2_\Omega \le Ch^{k+1}$. Here, we omit the proof of error estimates.

5 DG Methods for the Linear Schrödinger Equation

In this section, we consider the linear Schrödinger equation (1.5), and will formulate and analyze DG methods to solve it. Given the solution is complex-valued, throughout the section, the $L^2$ inner product and its induced norm

$$\begin{aligned} (w,v)_K=\int _K wv^*dK,\quad \Vert v\Vert _K=\sqrt{(v,v)_K}, \end{aligned}$$

(5.1)

are used for complex-valued square integrable functions in a domain K, and we will also work with the complex-valued discrete space $_c{\mathcal {V}}_h^k$, defined as

$$\begin{aligned} _c{\mathcal {V}}_h^k=\{v=r+is:r|_{I_j}\in P^k(I_j),s|_{I_j}\in P^k(I_j), \forall I_j\in {\mathcal {T}}_h\}. \end{aligned}$$

(5.2)

To obtain the DG methods for (1.5), we start with the first order form of the equation,

$$\begin{aligned} iu_t+p_x&=0,\quad p-u_x=0. \end{aligned}$$

(5.3)

Based on (5.3), our proposed DG method is to look for $u_h,p_h\in$$_c{\mathcal {V}}_h^k$ such that for any $v,w\in$$_c{\mathcal {V}}_h^k$ and for any j,

$$\begin{aligned} i\int _{I_j}(u_h)_tvdx-\int _{I_j}p_hv_xdx+(F_pv^-)_{j+\frac{1}{2}}-(F_pv^+)_{j-\frac{1}{2}}&=0, \end{aligned}$$

(5.4)

$$\begin{aligned} \int _{I_j}p_hwdx+\int _{I_j}u_hw_xdx-(F_uw^-)_{j+\frac{1}{2}}+(F_uw^+)_{j-\frac{1}{2}}&=0. \end{aligned}$$

(5.5)

Here $F_p$ and $F_u$ are numerical fluxes, taken as

$$\begin{aligned} F_p(p_h,u_h)=\{p_h\}+\alpha [p_h]+i\beta _1[u_h],\quad F_u(u_h,p_h)&=\{u_h\}-\alpha [u_h]+i\beta _2[p_h], \end{aligned}$$

(5.6)

and the parameters $\alpha ,\beta _1,\beta _2$ are O(1) and real-valued, and they will be specified later for stability and optimal accuracy. By summing up (5.4)–(5.5) over j, we obtain a compact form of the scheme: look for $u_h,p_h\in$$_c{\mathcal {V}}_h^k$ such that

$$\begin{aligned} B(u_h,p_h;v,w)=0,\quad \forall v, w\in {_c}{\mathcal {V}}^k_h, \end{aligned}$$

(5.7)

where

$$\begin{aligned} B(u_h,p_h;v,w)&=i\int _{\Omega }(u_h)_tvdx -\sum \limits _{j}\left( \int _{I_j}p_hv_xdx+(F_p[v])_{j-\frac{1}{2}}\right) \nonumber \\&\quad +\int _{\Omega }p_hwdx +\sum \limits _{j}\left( \int _{I_j}u_hw_xdx+(F_u[w])_{j-\frac{1}{2}}\right) . \end{aligned}$$

(5.8)

5.1 $L^2$ Stability

In this subsection, the $L^2$ stability is established for the DG method (5.7)–(5.8) with the numerical fluxes (5.6) under some assumptions on the parameters. The analysis relies on three energy equations.

$\bullet$The first energy equation First, we take the test functions $v=u_h^*$, $w=p_h^*$ in (5.7), and obtain

$$\begin{aligned} 0=B(u_h,p_h;u_h^*,p_h^*)&=i\int _{\Omega }(u_h)_tu_h^* dx -\sum \limits _{j}\left( \int _{I_j}p_h(u_h^*)_xdx+(F_p[u_h^*])_{j-\frac{1}{2}}\right) \nonumber \\&\quad +\int _{\Omega }p_hp_h^* dx +\sum \limits _{j}\left( \int _{I_j}u_h(p_h^*)_xdx+(F_u[p_h^*])_{j-\frac{1}{2}}\right) . \end{aligned}$$

(5.9)

We then subtract the conjugate of (5.9) from itself, and get

$$\begin{aligned} i\frac{d}{dt}\int _{\Omega }|u_h|^2dx+2i\text {Im}\left( \sum \limits _{j}\left( \int _{I_j}\left( p_h^*(u_h)_x+u_h(p_h^*)_x\right) dx+(F_u[p_h^*]+(F_p)^*[u_h])_{j-\frac{1}{2}}\right) \right) =0. \end{aligned}$$

This, together with the definition of the numerical fluxes in (5.6), leads to

$$\begin{aligned} \frac{d}{dt}\int _{\Omega }|u_h|^2dx+2\left( \sum \limits _{j}\left( -\beta _1[u_h][u_h^*]+\beta _2[p_h][p_h^*]\right) _{j-\frac{1}{2}}\right) =0. \end{aligned}$$

(5.10)

$\bullet$The second energy equation We here want to derive the energy equation for $p_h$. By taking the time derivative of (5.5), summing it up with (5.4) over j, we get

$$\begin{aligned} 0=B(u_h,(p_h)_t;v,w)&=i\int _{\Omega }(u_h)_tvdx -\sum \limits _{j}\left( \int _{I_j}p_hv_xdx+(F_p[v])_{j-\frac{1}{2}}\right) \nonumber \\&\quad +\int _{\Omega }(p_h)_twdx +\sum \limits _{j}\left( \int _{I_j}(u_h)_tw_xdx+((F_u)_t[w])_{j-\frac{1}{2}}\right) . \end{aligned}$$

(5.11)

With the test functions being $v=-(u_h^*)_t$ and $w=p_h^*$, (5.11) becomes

$$\begin{aligned} 0=B(u_h,(p_h)_t;-(u_h^*)_t,p_h^*)&=-i\int _{\Omega }(u_h)_t (u_h^*)_tdx +\sum \limits _{j}\left( \int _{I_j}p_h(u_h^*)_{tx}dx+(F_p[(u_h^*)_t])_{j-\frac{1}{2}}\right) \nonumber \\&\quad +\int _{\Omega }(p_h)_tp_h^* dx +\sum \limits _{j}\left( \int _{I_j}(u_h)_t(p_h^*)_xdx+((F_u)_t[p_h^*])_{j-\frac{1}{2}}\right) . \end{aligned}$$

(5.12)

Taking the conjugate of (5.12) and summing it up with (5.12), we have

$$\begin{aligned} 0&=\frac{d}{dt}\int _{\Omega }|p_h|^2dx+\sum \limits _{j}\left( \int _{I_j}(p_h(u_h^*)_{tx}+(u_h^*)_{t}(p_h)_x)dx+(F_p[(u_h)^*_t]+(F_u)^*_t[p_h])_{j-\frac{1}{2}}\right) \nonumber \\&\quad +\sum \limits _{j}\left( \int _{I_j}(p_h^*(u_h)_{tx}+(u_h)_t(p_h^*)_{x})dx+((F_p)^*[(u_h)_t]+(F_u)_t[p_h^*])_{j-\frac{1}{2}}\right) . \end{aligned}$$

(5.13)

Combining (5.13) with the definition of $F_p$, $F_u$ in (5.6), we have

$$\begin{aligned} \frac{d}{dt}\int _{\Omega }|p_h|^2dx+2\text {Im}\left( \sum \limits _{j}\left( -\beta _1[u_h][(u_h^*)_t]+\beta _2[(p_h^*)_t][p_h]\right) _{j-\frac{1}{2}}\right) =0. \end{aligned}$$

(5.14)

$\bullet$The third energy equation We start with taking the time derivative of (5.7), and then follow a similar procedure as to derive the first energy equation, except the test functions being taken as $v=(u_h^*)_t$, $w=(p_h^*)_t$. This leads to the third energy equation,

$$\begin{aligned} \frac{d}{dt}\int _{\Omega }|(u_h)_t|^2dx+2\sum \limits _{j}\left( -\beta _1[(u_h)_t][(u_h^*)_t]+\beta _2[(p_h)_t][(p_h^*)_t]\right) _{j-\frac{1}{2}}=0. \end{aligned}$$

(5.15)

By summing up the three energy equations (5.10), (5.14) and (5.15), we now have

$$\begin{aligned}&\frac{d}{dt}\int _{\Omega }\left( |u_h|^2+|p_h|^2+|(u_h)_t|^2\right) dx-2\beta _1\left( \sum \limits _{j}\left( |[u_h]|^2+|[(u_h)_t]|^2+\text {Im}([u_h][(u_h^*)_t])\right) _{j-\frac{1}{2}}\right) \nonumber \\&\quad +2\beta _2\left( \sum \limits _{j}\left( |[p_h]|^2+|[(p_h)_t]|^2+\text {Im}([(p_h^*)_t][p_h])\right) _{j-\frac{1}{2}}\right) =0, \end{aligned}$$

(5.16)

and this readily give us the $L^2$ stability result in the next Theorem.

Theorem 5.1

With $\beta _1\le 0$ and $\beta _2\ge 0$, the semi-discrete DG scheme (5.7)–(5.8) with the numerical fluxes (5.6) satisfies

$$\begin{aligned} \Vert u_h(T)\Vert ^2_\Omega +\Vert p_h(T)\Vert ^2_\Omega +\Vert (u_h)_t(T)\Vert ^2_\Omega \le \Vert u_h(0)\Vert ^2_\Omega +\Vert p_h(0)\Vert ^2_\Omega +\Vert (u_h)_t(0)\Vert ^2_\Omega . \end{aligned}$$

(5.17)

Remark 5.2

Compared with the Lemma 4.3 in [30], the proof for Theorem 5.1 requires an additional energy equation for $(u_h)_t$ with the presence of the parameters $\beta _1,\beta _2$. This energy relation is also important in error estimates to control both $p_h$ and $(u_h)_t$.

Remark 5.3

The proof for Theorem 5.1 is also different from the $L^2$ stability in Sect. 3.1.1 for the heat equation. Here we can not directly get the $L^2$ stability for $p_h$ unless we also have the energy relation for $(u_h)_t$.

5.2 $L^2$ Error Estimates

In this subsection, we will prove the optimal a prior$L^2$ error estimate for the DG method (5.7)–(5.8) for the linear Schrödinger equation (1.5) when the exact solution is sufficiently smooth, under the following assumption for the parameters in the numerical fluxes (5.6),

$$\begin{aligned} \alpha ^2-\beta _1\beta _2=\frac{1}{4}, \quad \beta _1\le 0,\; \beta _2\ge 0. \end{aligned}$$

(5.18)

Since the numerical fluxes (5.6) are consistent, the exact solution u and $p=u_x$ satisfy

$$\begin{aligned} B(u,p;v,w)=0,\quad \forall v,w\in {_c}{\mathcal {V}}^k_h, \end{aligned}$$

(5.19)

hence we get the error equation

$$\begin{aligned} B(e_u,e_p;v,w)=0,\quad \forall v,w\in {_c}{\mathcal {V}}^k_h. \end{aligned}$$

(5.20)

Here $e_\phi =\phi -\phi _h$, with $\phi =u, p$ are error functions. To ensure the error estimates to be optimal, we use a special projection,

$$\begin{aligned} _c\Pi _{h} \begin{pmatrix} u\\ p\\ \end{pmatrix} = \begin{pmatrix} P_h^+\left( (\frac{1}{2}-\alpha )u+i\beta _2p\right) +P_h^-\left( (\frac{1}{2}+\alpha )u-i\beta _2p\right) \\ P_h^+\left( (\frac{1}{2}+\alpha )p+i\beta _1u\right) +P_h^-\left( (\frac{1}{2}-\alpha )p-i\beta _1u\right) \end{pmatrix} \end{aligned}$$

(5.21)

that maps from ${_c}H^{1}(\Omega )\times {_c}H^{1}(\Omega )$ onto ${_c}{\mathcal {V}}^k_h\times {_c}{\mathcal {V}}^k_h$, $_cH^{1}(\Omega )$ denotes the function space with the real and the imaginary parts in $H^{1}(\Omega )$. Using $_c\Pi _h$, the error functions can be decomposed into $e_\phi =\eta _\phi +\zeta _\phi , \phi =u, p$ based on (2.10)–(2.11), with $\Pi _h$ replaced by $_c\Pi _h$.

The operator $_c\Pi _{h}$ is motivated by $\Pi _h$ in (2.9) (also see [9]), and it is tailored for the numerical flux (5.6). Following a similar proof for Lemma 2.4 in [9], we can show the following Lemma.

Lemma 5.4

For any given $\alpha$, $\beta _1$, $\beta _2$, the operator $_c\Pi _h$ has the following properties:

$$\begin{aligned} (i)&\quad \int _{I_j} \eta _u\phi _x^*dx=0, \quad \int _{I_j}\eta _p\psi _x^*dx=0, \quad \forall \phi ,\psi \in {_c}{\mathcal {V}}^k_h, \quad \forall j, \end{aligned}$$

(5.22)

$$\begin{aligned} (ii)&\quad \left\| \begin{pmatrix} u\\ p\\ \end{pmatrix} {-} {_c\Pi _h} \begin{pmatrix} u\\ p\\ \end{pmatrix} \right\| \le C_*\left( 1{+}|\alpha |{+}\max (|\beta _1|,|\beta _2|)\right) h^{k+1}\left( \Vert u\Vert _{H^{k+1}(\Omega )}{+}\Vert p\Vert _{H^{k+1}(\Omega )}\right) . \end{aligned}$$

(5.23)

If we further assume $\alpha ^2-\beta _1\beta _2=\frac{1}{4}$, we have

$$\begin{aligned} (iii)&\quad _c\Pi _h \text{ defines } \text{ a } \text{ projection, } \text{ that } \text{ is } (_c\Pi _h)^2= {_c\Pi _h}, \end{aligned}$$

(5.24)

$$\begin{aligned} (iv)&\quad F_p(\eta _p,\eta _u)_{j-\frac{1}{2}}=0,\quad F_u(\eta _u,\eta _p)_{j-\frac{1}{2}}=0, \quad \forall j. \end{aligned}$$

(5.25)

Here $F_p$ and $F_u$ are defined in (5.6), and $(u,p)\in {_c}H^{k+1}(\Omega )\times {_c}H^{k+1}(\Omega )$.

We choose the initial condition $p_h(x,0)=P_h^-p(x,0)$ with $p(x,0)=u_x(x,0)$. Using $p_h(x,0)$, we can further define the initial data $u_h(x, 0) \in {\mathcal {V}}_h^k$ which satisfies

$$\begin{aligned}&(p_h,w)_{I_j}+(u_h,w_x)|_{I_j}-({\widehat{u}}_hw^-)|_{j+\frac{1}{2}}+({\widehat{u}}_hw^+)|_{j-\frac{1}{2}}=0,\; \forall w\in {\mathcal {V}}_h^k,\; \nonumber \\&\quad \text{ with }\;{\widehat{u}}_h=u_h^+(x,0) \;\;\text{ and }\;\; \int _\Omega u_h(x,0)dx=\int _\Omega u(x,0)dx. \end{aligned}$$

(5.26)

Then, following the analysis for Lemma 5.1 in [23] and Lemma 2.4 in [30], we have the following estimates for the initial data.

Lemma 5.5

Assuming u(x, 0) is sufficiently smooth, the initial conditions described above are well defined and satisfy the following

$$\begin{aligned}&||p(x,0)-p_h(x,0)||_\Omega \le Ch^{k+1},\;||u(x,0)-u_h(x,0)||_\Omega \le Ch^{k+1},\nonumber \\&||u_t(x,0)-(u_h)_t(x,0)||_\Omega \le Ch^{k+1}. \end{aligned}$$

(5.27)

Here $(u_h)_t(x,0)$ is determined by (5.4) with $F_p=p_h^-(x,0)$ at $t=0$, and C depends on $||u(x,0)||_{H^{k+2}(\Omega )}$.

To obtain the optimal error estimates, we follow the line of stability analysis and get three error equations.

5.2.1 The First Error Equation

Taking the test functions $v=\zeta _u^*$ and $w=\zeta _p^*$ in the error equation (5.20), we have

$$\begin{aligned} B(\zeta _u,\zeta _p;\zeta _u^*,\zeta _p^*)+B(\eta _u,\eta _p;\zeta _u^*,\zeta _p^*)=0. \end{aligned}$$

(5.28)

Now we follow a similar procedure to get the first energy equation (5.9), and use the definition of B in (5.8), and get

$$\begin{aligned} B(\zeta _u,\zeta _p;\zeta _u^*,\zeta _p^*)&=i\int _{\Omega }(\zeta _u)_t\zeta _u^* dx -\sum \limits _{j}\left( \int _{I_j}\zeta _p(\zeta _u^*)_xdx+(F_p[\zeta _u^*])_{j-\frac{1}{2}}\right) \nonumber \\&+\int _{\Omega }\zeta _p\zeta _p^* dx +\sum \limits _{j}\left( \int _{I_j}\zeta _u(\zeta _p^*)_xdx+(F_u[\zeta _p^*])_{j-\frac{1}{2}}\right) , \end{aligned}$$

(5.29)

$$\begin{aligned} B(\eta _u,\eta _p;\zeta _u^*,\zeta _p^*)&=i\int _{\Omega }(\eta _u)_t\zeta _u^*dx+\int _{\Omega }\eta _p\zeta _p^*dx. \end{aligned}$$

(5.30)

Here, we have used the properties (5.22) and (5.25) of the projection ${_c\Pi }_h$ for (5.30). Now we subtract (5.28) by its conjugate, and use (5.29) and (5.30), this will lead to

$$\begin{aligned}&\frac{d}{dt}\int _{\Omega }|\zeta _u|^2dx+2\left( \sum \limits _{j}\left( -\beta _1[\zeta _u][\zeta _u^*]+\beta _2[\zeta _p][\zeta _p^*]\right) _{j-\frac{1}{2}}\right) \nonumber \\&\quad =-2\text {Re}\left( \int _{\Omega }(\eta _u)_t\zeta _u^*dx\right) -2\text {Im}\left( \int _{\Omega }\eta _p\zeta _p^*dx\right) . \end{aligned}$$

(5.31)

5.2.2 The Second Error Equation

Following the second energy equation (5.11) and replacing $u_h,p_h$ by $e_u,e_p$, we have

$$\begin{aligned} B(\zeta _u,(\zeta _p)_t;v,w)+B(\eta _u,(\eta _p)_t;v,w)=0,\quad \forall v,w\in {_c}{\mathcal {V}}^k_h. \end{aligned}$$

(5.32)

Taking the test functions $v=-(\zeta _u^*)_t$ and $w=\zeta _p^*$ in (5.32), we get

$$\begin{aligned} B(\zeta _u,(\zeta _p)_t;-(\zeta _u^*)_t,\zeta _p^*)=-B(\eta _u,(\eta _p)_t;-(\zeta _u^*)_t,\zeta _p^*), \end{aligned}$$

(5.33)

with

$$\begin{aligned} B(\zeta _u,(\zeta _p)_t;-(\zeta _u^*)_t,\zeta _p^*)&=-i\int _{\Omega }(\zeta _u)_t(\zeta _u^*)_t dx +\sum \limits _{j}\left( \int _{I_j}\zeta _p(\zeta _u^*)_{tx}dx+(F_p[(\zeta _u^*)_t])_{j-\frac{1}{2}}\right) \nonumber \\&\quad +\int _{\Omega }(\zeta _p)_t\zeta _p^* dx +\sum \limits _{j}\left( \int _{I_j}(\zeta _u)_t(\zeta _p^*)_xdx+((F_u)_t[\zeta _p^*])_{j-\frac{1}{2}}\right) ,\end{aligned}$$

(5.34)

$$\begin{aligned} B(\eta _u,(\eta _p)_t;-(\zeta _u^*)_t,\zeta _p^*)&=-i\int _{\Omega }(\eta _u)_t(\zeta _u^*)_tdx+\int _{\Omega }(\eta _p)_t\zeta _p^*dx. \end{aligned}$$

(5.35)

We now add (5.33) and its conjugate and get

$$\begin{aligned}&\frac{d}{dt}\int _{\Omega }|\zeta _p|^2dx+2\text {Im}\left( \sum \limits _{j}\left( -\beta _1[\zeta _u][(\zeta _u^*)_t]+\beta _2[(\zeta _p^*)_t][\zeta _p]\right) _{j-\frac{1}{2}}\right) \nonumber \\&\quad =-2\text {Im}\int _{\Omega }(\eta _u)_t(\zeta _u^*)_tdx-2\text {Re}\int _{\Omega }(\eta _p)_t\zeta _p^*dx. \end{aligned}$$

(5.36)

Here, the definition of $F_p,F_u$ in (5.6) and the properties (5.22), (5.25) of the projection ${_c\Pi }_h$ are used.

5.2.3 The Third Error Equation

In the last step, we follow the third energy equation (5.15). Similar to the equation (5.28), the test functions are taken to be $v=(\zeta _u^*)_t,w=(\zeta _p^*)_t$. Then, we have

$$\begin{aligned} B((\zeta _u)_t,(\zeta _p)_t;(\zeta _u^*)_t,(\zeta _p^*)_t)=-B((\eta _u)_t,(\eta _p)_t;(\zeta _u^*)_t,(\zeta _p^*)_t). \end{aligned}$$

(5.37)

We subtract (5.37) by its conjugate, and get

$$\begin{aligned}&\frac{d}{dt}\int _{\Omega }|(\zeta _u)_t|^2dx+2\left( \sum \limits _{j}\left( -\beta _1[(\zeta _u)_t][(\zeta _u^*)_t]+\beta _2[(\zeta _p)_t][(\zeta _p^*)_t]\right) _{j-\frac{1}{2}}\right) \nonumber \\&\quad =-2\text {Re}\left( \int _{\Omega }(\eta _u)_{tt}(\zeta _u^*)_tdx\right) -2\text {Im}\left( \int _{\Omega }(\eta _p)_t(\zeta _p^*)_tdx\right) . \end{aligned}$$

(5.38)

Once we have the three error equations, (5.31), (5.36) and (5.38), we sum them up and get

$$\begin{aligned}&\frac{d}{dt}\int _{\Omega }\left( |(\zeta _u)_t|^2+|\zeta _p|^2+|\zeta _u|^2\right) dx\nonumber \\&\qquad -2\beta _1\sum \limits _{j}\left( |[\zeta _u]|^2+\text {Im}([\zeta _u][(\zeta _u^*)_t])+|[(\zeta _u)_t]|^2\right) _{j-\frac{1}{2}}\nonumber \\&\qquad +2\beta _2\sum \limits _{j}\left( |[\zeta _p]|^2+\text {Im}([\zeta _p][(\zeta _p^*)_t])+|[(\zeta _p)_t]|^2\right) _{j-\frac{1}{2}}\nonumber \\&\quad =2\text {Re}(\Theta )+2\text {Im}(\Gamma )+2\text {Im}(\Lambda ). \end{aligned}$$

(5.39)

Here, $\Theta ,\Gamma ,\Lambda$ denote

$$\begin{aligned} \Theta&=-\int _{\Omega }\left( (\eta _u)_t\zeta _u^*+(\eta _p)_t\zeta _p^*+(\eta _u)_{tt}(\zeta _u^*)_t\right) dx, \end{aligned}$$

(5.40)

$$\begin{aligned} \Gamma&=-\int _{\Omega }\left( \eta _p\zeta _p^*+(\eta _u)_t(\zeta _u^*)_t\right) dx,\quad \Lambda =-\int _{\Omega }(\eta _p)_t(\zeta _p^*)_tdx. \end{aligned}$$

(5.41)

Related to $\Lambda$, we have

$$\begin{aligned} \int _0^T\Lambda dt =&-\int _{\Omega }( (\eta _p)_{t}\zeta _p^*)dx|_0^T+\int _0^T\int _{\Omega }(\eta _p)_{tt}\zeta _p^*dt, \end{aligned}$$

(5.42)

hence

$$\begin{aligned} \int _0^T\text {Im}(\Lambda )dt\le&Ch^{2k+2}+\frac{1}{4}\int _{\Omega }|\zeta _p|^2dx+\frac{1}{8}\int _0^T\int _{\Omega }|\zeta _p|^2dxdt. \end{aligned}$$

(5.43)

As for $\Theta$ and $\Gamma$, we have

$$\begin{aligned} \text {Re}(\Theta )&\le Ch^{2k+2}+\frac{1}{8}\int _{\Omega }(2|\zeta _u|^2+|(\zeta _u)_t|^2+\frac{1}{2}|\zeta _p|^2)dx,\nonumber \\ \text {Im}(\Gamma )&\le Ch^{2k+2}+\frac{1}{8}\int _{\Omega }(|(\zeta _u)_t|^2+\frac{1}{2}|\zeta _p|^2), \end{aligned}$$

(5.44)

thus

$$\begin{aligned} \int _0^T(\text {Re}(\Theta )+\text {Im}(\Gamma ))dt\le Ch^{2k+2} + \frac{1}{4}\int _0^T\int _{\Omega }\left( |\zeta _u|^2+|(\zeta _u)_t|^2+\frac{1}{2}|\zeta _p|^2\right) dxdt. \end{aligned}$$

(5.45)

Here, we have used the Young inequality and the optimal error estimates in Lemma 5.5 from the initialization. Now we combine (5.39)–(5.45) with $\beta _1\le 0$ and $\beta _2\ge 0$, and get

$$\begin{aligned} \int _{\Omega }\left( |(\zeta _u)_t|^2+\frac{1}{2}|\zeta _p|^2+|\zeta _u|^2\right) |_{t=T}dx&\le Ch^{2k+2} \nonumber \\&\quad + \frac{1}{2}\int _0^T\int _{\Omega }\left( |\zeta _u|^2+|(\zeta _u)_t|^2+\frac{1}{2}|\zeta _p|^2\right) dxdt. \end{aligned}$$

(5.46)

In this section, C depends on k, T, $\Vert u\Vert _{L^{\infty }((0,T);H^{k+2}(\Omega ))}$, $\Vert u_t\Vert _{L^{\infty }((0,T);H^{k+2}(\Omega ))}$ and $\Vert u_{tt}\Vert _{L^{\infty }((0,T);H^{k+2}(\Omega ))}$. Finally we apply the Gronwall’s inequality to (5.46), and reach the following theorem.

Theorem 5.6

For the semi-discrete DG scheme (5.7)–(5.8) with the numerical fluxes (5.6) under the conditions in (5.18), the following error estimate holds when the exact solution u of the Eq. (1.5) is sufficiently smooth,

$$\begin{aligned} \Vert e_u\Vert {^2_\Omega }+\Vert (e_u)_t\Vert {^2_\Omega }+\Vert e_p\Vert {^2_\Omega }\le Ch^{2k+2}. \end{aligned}$$

(5.47)

Here $p=u_x$, and C depends on $\Vert u\Vert _{L^{\infty }((0,T);H^{k+2}(\Omega ))}$, $\Vert u_t\Vert _{L^{\infty }((0,T);H^{k+2}(\Omega ))}$ and $\Vert u_{tt}\Vert _{L^{\infty }((0,T);H^{k+2}(\Omega ))}$.

6 Numerical Examples

In this section, we present numerical examples to demonstrate the performance of the proposed methods, and to verify our theoretical results in previous sections. In our numerical experiments, the implicit $(k+1)$-th order SDC method in [22, 28] is utilized as the time discretization for the DG methods when the $P^k$ polynomial spaces are used. The implicit SDC temporal discretization allows the time step to be $\Delta t=O(h)$ and can be easily implemented to have arbitrary order of accuracy. In all numerical tests, uniform meshes with N cells are used.

6.1 The Heat Equation

In this subsection, we consider the one dimensional heat equation (1.1), with the initial data

$$\begin{aligned} u(x,0)&=\sin (x), \quad \text {on} \quad (0,2\pi ), \end{aligned}$$

(6.1)

and the periodic boundary condition. The exact solution is $u(x,t)=e^{-t}\sin (x)$. We test the problem with different sets of $\alpha ,\beta _1, \beta _2$, and compute up to $T=5$ based on $P^k$ approximations with $k=1,2,3$. The time step is taken as $\Delta t=h$. Tables 1 and 2 show $L^2$ errors and orders of accuracy of the numerical solution $u_h$ and the auxiliary variable $p_h$. We observe that both $u_h$ and $p_h$ from the proposed DG methods for the Eq. (1.1) are $(k+1)$-th order accurate with $k=1,2,3$ when the numerical fluxes (3.4) satisfy the conditions, $\alpha ^2+\beta _1\beta _2=\frac{1}{4}, \beta _1\ge 0, \beta _2\ge 0$, and this verifies our theoretical results. On the other hand, when the numerical fluxes are central (with $\alpha =\beta _1=\beta _2=0$), and it does not satisfies $\alpha ^2+\beta _1\beta _2=\frac{1}{4}$, the numerical solutions are suboptimal when k is odd.

Table 1 $L^2$ errors and orders of accuracy of $u_h$ for the heat equation (1.1) at time $T=5$

Table 2 $L^2$ errors and orders of accuracy of $p_h$ with $p=u_x$ for the heat equation (1.1) at time $T=5$

6.2 The Fourth Order Equation

Here, we consider the fourth order equation (1.2) with the initial condition

$$\begin{aligned} u(x,0)&=\sin (x), \quad \text {on} \quad (0,2\pi ). \end{aligned}$$

(6.2)

Periodic boundary condition is used. The exact solution is $u(x,t)=e^{-t}\sin (x)$. Several sets of the parameters $\alpha , \beta _1,\beta _2$ are used in the numerical fluxes (3.28) to test the proposed DG methods, and the problem is computed up to $T=5$ based on $P^k$ approximation with $k=1,2,3$. For this test, we use the time step $\Delta t=h$. The numerical results are shown in Table 3 for the $L^2$ errors and orders of accuracy of the numerical solution $u_h$, and they confirm the $(k+1)$-th order of accuracy for u with $k=1,2,3$, when the parameter set satisfies $\alpha ^2+\beta _1\beta _2=\frac{1}{4}, \beta _1\ge 0, \beta _2\ge 0$. The DG method with the central flux, which does not satisfy the condition above, yields the suboptimal rate when k is odd.

In Table 4, we present the $L^2$ errors and orders of accuracy of the auxiliary variables $r_h,q_h,p_h$ with $r=u_x,q=r_x,p=q_x$ from the DG methods which use the numerical fluxes (3.28) with $\alpha =0.499,\beta _1=\beta _2=\sqrt{\frac{1}{4}-\alpha ^2}$ and $\alpha =-0.5,\beta _1=0,\beta _2=0.5$. The auxiliary variables are optimally accurate for these choices. When the remaining parameter choices from Table 3 are used, similar observations as for $u_h$ are observed for the auxiliary variables in terms of the convergence orders, and the results are not reported here.

Table 3 $L^2$ errors and orders of accuracy of $u_h$ for the fourth order equation (1.2) at time $T=5$

Table 4 $L^2$ errors and orders of accuracy of auxiliary variables $p_h$, $q_h$, $r_h$ for the fourth order equation (1.2) at time $T=5$

Table 5 $L^2$ errors and orders of accuracy of $u_h$ for the third order wave equation (1.4) at time $T=5$

6.3 The Third Order Wave Equation

In this subsection, we test the third order wave equation (1.4) with the initial condition

$$\begin{aligned} u(x,0)&=\sin (x), \quad \text {on} \quad (0,2\pi ), \end{aligned}$$

(6.3)

and the periodic boundary condition. The exact solution is $u(x,t)=\sin (x+t)$. We test the problem with several sets of the parameters $\alpha , \beta _1,\beta _2$, and compute the problem up to time $T=5$ based on $P^k$ approximations with $k=1,2,3$. The time step is taken as $\Delta t=h$. In Table 5, we report the results for the $L^2$ errors and orders of accuracy of the numerical solution $u_h$. From these results, we see that the numerical solution $u_h$ is optimal i.e $(k+1)$-th order with $k=1,2,3$, when $\alpha , \beta _1,\beta _2$ satisfy the flux conditions in (4.18). We also note that the central flux, which does not satisfies $\alpha ^2+\beta _1\beta _2=\frac{1}{4}$, yields the suboptimal rate when k is odd. In Table 6, we present the $L^2$ errors and orders for the auxiliary variables $q_h$, $p_h$ with $q=u_x,p=q_x$, using the parameters $\alpha =-0.499, \beta _1=\beta _2=-\sqrt{0.25-0.499^2}$, $\alpha =-0.5,\beta _1=0,\beta _2=-0.5$ and $\alpha =\beta _1=\beta _2=0$. We observe that the auxiliary variables $p_h$, $q_h$ have the same accuracy as $u_h$, with one exception when the central fluxes are used for $u_h$ and $p_h$, and $F_q=q_h^+$ for $q_h$. In this case, $q_h$ is always optimal, while $u_h$ and $p_h$ are suboptimal when k is odd.

Table 6 $L^2$ errors and orders of accuracy of auxiliary variables $p_h$, $q_h$ for the third order wave equation (1.4) at time $T=5$

6.4 The Linear Schrödinger Equation

Finally, we test the one dimensional linear Schrödinger problem (1.5) with the initial data

$$\begin{aligned} u(x,0)&=\sin (x), \quad \text {on} \quad (0,2\pi ), \end{aligned}$$

(6.4)

and with the periodic boundary condition. The exact solution is $u(x,t)=e^{i(x-t)}$. We test this problem with different sets of $\alpha ,\beta _1, \beta _2$, and compute up to $T=5$ based on $P^k$ approximations with $k=1,2,3$. In our test for this problem, the CFL constraint for the method with central flux $(\alpha =\beta _1=\beta _2=0)$ is taken to be 0.2 and 1.0 for other choices of parameters in numerical fluxes. We show the numerical results in Tables 7 and 8 for $L^2$ errors and orders of accuracy of the numerical solutions $u_h$ and the auxiliary variable $p_h$. From these results, we see both the numerical solution $u_h$ and auxiliary variable $p_h$ have the optimal accuracy with $k=1,2,3$ under the conditions $\alpha ^2-\beta _1\beta _2=\frac{1}{4},\beta _1\le 0,\beta _2\ge 0$. On the other hand, the DG method with the central flux (with $\alpha =\beta _1=\beta _2=0$) yields the suboptimal accuracy.

Table 7 $L^2$ errors and orders of accuracy for $u_h$ for the linear Schrödinger equation (1.5) at $T=5$

Table 8 $L^2$ errors and orders of accuracy for $p_h$ with $p=u_x$ for the linear Schrödinger equation (1.5) at $T=5$

7 Conclusion

In this paper, we have developed DG methods for solving the even-order equations (including the heat and a fourth order equation), a third order wave equation, and the linear Schrödinger equation in one dimension. A general class of numerical fluxes is identified to ensure the optimal accuracy of the numerical solution and of some auxiliary variables. A set of energy relations, as well as the design of special projection operators are the key to achieve the optimality of the error estimates. In future work, we want to extend the study to high dimensions and to nonlinear models.

Appendices

A The Proof of Theorem 4.3

In this appendix, we prove the $L^2$ stability of the DG scheme (4.6)–(4.7) with the numerical fluxes (4.5) and $\alpha =\frac{1}{2},\beta _1=\beta _2=0$, namely

$$\begin{aligned} F_p =p_h^+, \quad F_q = q_h^+, \quad F_u=u_h^-. \end{aligned}$$

(A.1)

Five energy equations will be derived first.

$\bullet$The first energy equation. With $\alpha =\frac{1}{2},\beta _1=\beta _2=0$, the first energy equation (4.9) becomes

$$\begin{aligned} 0=B(u_h,p_h,q_h;u_h,q_h,-p_h)=\frac{1}{2}\frac{d}{dt}\int _{\Omega }u_h^2dx +\frac{1}{2}\sum \limits _{j}[q_h]^2_{j-\frac{1}{2}}. \end{aligned}$$

(A.2)

$\bullet$The second energy equation. We take the test functions $v=-\frac{1}{2}(q_h)_t$, $w=\frac{1}{2}p_h$ and $z=0$ in (4.10), use the definition of $F_p$, $F_q$ and $F_u$ in (A.1), and obtain

$$\begin{aligned} 0&=B(u_h,(p_h)_t,(q_h)_t;-\frac{1}{2}(q_h)_t,\frac{1}{2}p_h,0)\nonumber \\&=-\frac{1}{2}\int _{\Omega }(u_h)_t(q_h)_t dx + \frac{1}{2}\sum \limits _{j}\left( \int _{I_j}p_h (q_h)_{tx}dx+(F_p[(q_h)_t])_{j-\frac{1}{2}}\right) \nonumber \\&\quad +\frac{1}{2}\int _{\Omega }(p_h)_t p_h dx + \frac{1}{2}\sum \limits _{j}\left( \int _{I_j}(q_h)_t(p_h)_xdx+((F_q)_t[p_h])_{j-\frac{1}{2}}\right) \nonumber \\&=\frac{1}{2}\int _{\Omega }(-(u_h)_t(q_h)_t+(p_h)_tp_h)dx+\frac{1}{2}\sum \limits _{j}\left( [p_h][(q_h)_t]\right) _{j-\frac{1}{2}}. \end{aligned}$$

(A.3)

$\bullet$The third energy equation. With $\alpha =\frac{1}{2},\beta _1=\beta _2=0$, (4.14) becomes

$$\begin{aligned} 0=&B((u_h)_t,(p_h)_t,(q_h)_t;(u_h)_t,(q_h)_t,-(p_h)_t)=\frac{1}{2}\frac{d}{dt}\int _{\Omega }(u_h)_t^2dx +\frac{1}{2}\sum \limits _{j}[(q_h)_t]^2_{j-\frac{1}{2}}. \end{aligned}$$

(A.4)

$\bullet$The fourth energy equation. Here we take the test functions $v=-p_h$, $w=(u_h)_t$ and $z=q_h$ in (4.15), use the definition of $F_p$, $F_q$ and $F_u$ in (A.1), and obtain

$$\begin{aligned} 0&=B\left( u_h,p_h,\left( q_h\right) _t;-p_h,\left( u_h\right) _t,q_h\right) \nonumber \\&=-\int _{\Omega }\left( u_h\right) _t p_h dx +\sum \limits _{j}\left( \int _{I_j}p_h\left( p_h\right) _{x}dx+\left( F_p[p_h]\right) _{j-\frac{1}{2}}\right) \nonumber \\&\quad +\int _{\Omega }p_h \left( u_h\right) _t dx + \sum \limits _{j}\left( \int _{I_j}q_h\left( u_h\right) _{tx}dx+\left( F_q[\left( u_h\right) _t]\right) _{j-\frac{1}{2}}\right) \nonumber \\&\quad + \int _{\Omega }\left( q_h\right) _tq_h dx + \sum \limits _{j}\left( \int _{I_j}\left( u_h\right) _t \left( q_h\right) _xdx+\left( \left( F_u\right) _t[q_h]\right) _{j-\frac{1}{2}}\right) \nonumber \\&=\frac{1}{2}\frac{d}{dt} \int _{\Omega }\left( q_h\right) ^2 dx+\frac{1}{2}\sum \limits _{j}[p_h]^2_{j-\frac{1}{2}}. \end{aligned}$$

(A.5)

$\bullet$The fifth energy equation. We take the time derivative of (4.15) and get

$$\begin{aligned} B((u_h)_t,(p_h)_t,(q_h)_{tt};v,w,z)=0, \quad \forall v,w,z\in {\mathcal {V}}_h^k. \end{aligned}$$

(A.6)

With the test functions in (A.6) taken as $v=-(p_h)_t$, $w=(u_h)_{tt}$ and $z=(q_h)_t$, using the definition of $F_p$, $F_q$ and $F_u$ in (A.1), we have

$$\begin{aligned} 0=B((u_h)_t,(p_h)_t,(q_h)_{tt};-(p_h)_t,(u_h)_{tt},(q_h)_t)=\int _{\Omega }(q_h)_{tt}(q_h)_t dx+\frac{1}{2}\sum \limits _{j}[(p_h)_t]^2_{j-\frac{1}{2}}. \end{aligned}$$

(A.7)

$\bullet$Proof of Theorem4.3

Proof

We sum up the energy equations (A.2)–(A.5) and (A.7), and get

$$\begin{aligned} 0&=\frac{1}{2}\frac{d}{dt}\left( \Vert u_h\Vert ^2_\Omega +\frac{1}{2}\Vert p_h\Vert ^2_\Omega +\Vert (u_h)_t\Vert ^2_\Omega +\Vert q_h\Vert ^2_\Omega +\Vert (q_h)_t\Vert ^2_\Omega \right) -\frac{1}{2}((q_h)_t,(u_h)_t)\nonumber \\&\quad +\frac{1}{2}\sum \limits _{j}\left( [q_h]^2+[(q_h)_t]^2+[(p_h)_t]^2+[p_h]^2+[p_h][(q_h)_t]\right) _{j-\frac{1}{2}}. \end{aligned}$$

(A.8)

Thus, we have

$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\left( \Vert u_h\Vert ^2_\Omega +\frac{1}{2}\Vert p_h\Vert ^2_\Omega +\Vert (u_h)_t\Vert ^2_\Omega +\Vert q_h\Vert ^2_\Omega +\Vert (q_h)_t\Vert ^2_\Omega \right) \le \frac{1}{4}(\Vert (q_h)_t\Vert ^2_\Omega +\Vert (u_h)_t\Vert ^2_\Omega ). \end{aligned}$$

(A.9)

We integrate (A.9) with respect to time over [0, T], and obtain

$$\begin{aligned}&\Vert u_h(T)\Vert ^2_\Omega +\frac{1}{2}\Vert p_h(T)\Vert ^2_\Omega +\Vert (u_h)_t(T)\Vert ^2_\Omega +\Vert q_h(T)\Vert ^2_\Omega +\Vert (q_h)_t(T)\Vert ^2_\Omega \nonumber \\&\quad \le \frac{1}{2}\int _0^T(\Vert (q_h)_t(t) \Vert ^2_\Omega +\Vert (u_h)_t(t) \Vert ^2_\Omega )dt\nonumber \\&\qquad +\left( \Vert u_h(0)\Vert ^2_\Omega +\frac{1}{2}\Vert p_h(0)\Vert ^2_\Omega +\Vert (u_h)_t(0)\Vert ^2_\Omega +\Vert q_h(0)\Vert ^2_\Omega +\Vert (q_h)_t(0)\Vert ^2_\Omega \right) . \end{aligned}$$

(A.10)

From the third energy equation (A.4) and the fifth energy equation (A.7), we have

$$\begin{aligned} \Vert (u_h)_t(t)\Vert ^2_\Omega \le \Vert (u_h)_t(0)\Vert ^2_\Omega , \quad \Vert (q_h)_t(t)\Vert ^2_\Omega \le \Vert (q_h)_t(0)\Vert ^2_\Omega , \quad \forall t\ge 0. \end{aligned}$$

(A.11)

Therefore, we can obtain the $L^2$ stability in Theorem 4.3. $\square$

B The Derivation of the Conditions (4.18)

In this appendix, we will give the derivation of the conditions (4.18) that are to ensure the matrices $S_1$ and $S_2$ in (4.22) to be positive semi-definite. For this, we use the following sufficient and necessary condition for an $n\times n$ matrix to be positive semi-definite: all the principal minors$D_k$are nonnegative, $k=1, \ldots n$. Here, $D_k$ is formed by deleting any$n-k$ rows and the corresponding columns. Additionally, we require the relation $\alpha ^2+\beta _1\beta _2=\frac{1}{4}$, which helps with simplifying the conditions and is also needed for optimal accuracy.

For the matrix $S_1$, the first principal minors are

$$\begin{aligned} D_{1,1}=-\beta _1,\;D_{1,2}=-\beta _2,\;D_{1,3}=\frac{1}{2}; \end{aligned}$$

(B.1)

the second principal minors are

$$\begin{aligned} D_{2,1}=\begin{vmatrix} -\beta _2&\frac{\alpha +0.5}{2} \\ \frac{\alpha +0.5}{2}&\frac{1}{2} \end{vmatrix} ,\; D_{2,2}=\begin{vmatrix} -\beta _1&\frac{\beta _1}{2} \\ \frac{\beta _1}{2}&\frac{1}{2} \end{vmatrix} ,\; D_{2,3}=\begin{vmatrix} -\beta _1&0 \\ 0&-\beta _2 \end{vmatrix} ; \end{aligned}$$

(B.2)

and the third principal minor is

$$\begin{aligned} D_3=|S_1|=\frac{1}{2}\beta _1\beta _2+\frac{1}{4}\beta _1^2\beta _2+\frac{1}{4}\beta _1{\left( \alpha +\frac{1}{2}\right) ^2}. \end{aligned}$$

(B.3)

Let the first principal minors be nonnegative, we have $\beta _1\le 0$ and $\beta _2\le 0$. From the second principal minors being nonnegative, we obtain

$$\begin{aligned} 2\beta _2+\left( \alpha +\frac{1}{2}\right) ^2\le 0,\;\;\beta _1(2+\beta _1)\le 0,\; \;\beta _1\beta _2\ge 0. \end{aligned}$$

(B.4)

Let the third principal minor $D_3$ be nonnegative, with $\beta _1\le 0$ and the assumption $\alpha ^2+\beta _1\beta _2=\frac{1}{4}$, we get

$$\begin{aligned} 2\beta _2+\beta _1\beta _2+(\alpha +\frac{1}{2})^2 =2\beta _2+\alpha +\frac{1}{2}\le 0. \end{aligned}$$

(B.5)

We observe that, with $\beta _1\beta _2\ge 0$, the inequality (B.5) will automatically ensure the first inequality in (B.4). Combining all we have so far, the following conditions are derived to ensure $S_1$ be positive semi-definite

$$\begin{aligned} -2\le \beta _1\le 0,\;\;\beta _2\le 0,\;\;2\beta _2+\alpha \le -\frac{1}{2},\;\;\alpha ^2+\beta _1\beta _2=\frac{1}{4}. \end{aligned}$$

(B.6)

Fig. 1

The region of $\beta _1,\beta _2$ in the condition (4.18) for the stability. Left: $\alpha =-\sqrt{\frac{1}{4}-\beta _1\beta _2}$; Right: $\alpha =-\frac{1}{4}$

For the matrix $S_2$, we follow the similar analysis as for $S_1$. By requiring all the first and the second principal minors being nonnegative, we get

$$\begin{aligned}&\beta _1+\alpha \le 0, \quad \beta _2\le 0, \end{aligned}$$

(B.7)

$$\begin{aligned}&8(\beta _1+\alpha )+\left( \frac{1}{2}-\alpha \right) ^2\le 0,\; \; \beta _2(8+\beta _2)\le 0,\; \;(\alpha +\beta _1)\beta _2-\frac{1}{4}\beta _2^2\ge 0, \end{aligned}$$

(B.8)

Let the third order principal minor of $S_2$ be nonnegative, also with $\beta _2\ge 0$, we obtain

$$\begin{aligned}&8(\beta _1+\alpha )+\left( \frac{1}{2}-\alpha \right) ^2+\beta _2\left( \frac{1}{2}-\alpha \right) +\beta _2(\beta _1+\alpha )-2\beta _2\nonumber \\&\quad =8(\beta _1+\alpha )+\left( \frac{1}{2}-\alpha \right) ^2+\beta _2(\beta _1-\frac{3}{2}) \le 0. \end{aligned}$$

(B.9)

Using $\beta _2\le 0$ and assuming $\beta _1\le 0$, one can see that (B.9) implies the first inequality in (B.8), which on the other hand ensures the first inequality in (B.7). Combining (B.7)–(B.9) with $\alpha ^2+\beta _1\beta _2=\frac{1}{4}$, we have the conditions for $S_2$ as

$$\begin{aligned} \alpha ^2+\beta _1\beta _2=\frac{1}{4},\;\beta _1\le 0,\;-8\le \beta _2\le 0,\;4(\beta _1+\alpha )\le \beta _2,\;8\beta _1+7\alpha -\frac{3}{2}\beta _2\le -\frac{1}{2}. \end{aligned}$$

(B.10)

Finally, we reach the conditions in (4.18) by putting (B.6) and (B.10) together. To get some idea about these conditions in (4.18), we present two plots in Fig. 1. In the left figure, we plot those pairs $(\beta _1,\beta _2)$ such that with the respective $\alpha =-\sqrt{\frac{1}{4}-\beta _1\beta _2}$, the conditions in (4.18) are all satisfied. In the right figure, we fix $\alpha =-\frac{1}{4}$, and plot $(\beta _1,\beta _2)$ satisfying the conditions in (4.18). Note that such $(\beta _1,\beta _2)$ form part of the parabola: $\beta _1\beta _2=\frac{1}{4}-\alpha ^2=\frac{3}{16}$, see the solid line in red.