A High-Order Discontinuous Galerkin Lagrange Projection Scheme for the Barotropic Euler Equations

Abstract

This work considers the barotropic Euler equations and proposes a high-order conservative scheme based on a Lagrange-Projection decomposition. The high-order in space and time are achieved using Discontinuous Galerkin (DG) and Runge-Kutta (RK) strategies. The use of a Lagrange-Projection decomposition enables the use of time steps that are not constrained by the sound speed thanks to an implicit treatment of the acoustic waves (Lagrange step), while the transport waves (Projection step) are treated explicitly. We compare our DG discretization with the recent one (Renac in Numer Math 1-27, 2016, [7]) and state that it satisfies important non linear stability properties. The behaviour of our scheme is illustrated by several test cases.

1 Introduction

We are interested in the gas dynamics equations in Eulerian coordinates

$$\begin{aligned} \left\{ \begin{aligned}&\partial _t \rho + \partial _x(\rho u) = 0, \\&\partial _t (\rho u)+\partial _x \left( \rho u^2 + p\right) = 0, \end{aligned} \right. \end{aligned}$$

(1)

where $\rho > 0$ is the density, u the velocity and $p=p(\rho )$ is the pressure of the fluid such that $p'(\rho ) > 0$. In the numerical experiments, we will choose $p(\rho )=g \rho ^2/2$ where $g > 0$ is the gravity constant so that the model can also be understood as the Shallow-Water equations with flat topography (in this case, $\rho$ stands for the water depth). The unknowns depend on the space and time variables x and t, with $x \in \mathbb {R}$ and $t \in [0,\infty )$. At time $t=0$, the model is supplemented with a given initial data $\rho (x,t=0)~=~\rho _0(x)$ and $u(x,t=0)~=~u_0(x)$.

The aim of this paper is to propose a high-order discretization based on a Lagrange-Projection decomposition of the governing equations and using a Discontinuous Galerkin (DG) [4, 9] strategy for the space variable.

The Lagrange-Projection (or equivalently Lagrange-Remap) decomposition is interesting since it allows to naturally decouple the acoustic and transport terms of the model. It proved to be useful and very efficient when considering subsonic or low-Mach number flows. In this case, the CFL restriction of Godunov-type schemes is driven by the acoustic waves and can be very restrictive. The Lagrange-Projection strategy allows for a very natural implicit-explicit scheme with a CFL restriction based on the (slow) transport waves and not on the (fast) acoustic waves. We refer for instance the reader to [1, 2, 5], to the recent contribution [3], and to the references therein. Note that the later contribution considers the Shallow-Water equations with non flat topography and that the corresponding (implicit-explicit) Lagrange-Projection scheme is well-balanced but only first-order accurate. It is the purpose of this contribution to extend the first-order Lagrange-Projection schemes of the above references to high-order of accuracy in both space and time. The proposed approach is quite close to the one recently developed in [7], but as we will see, the corresponding Projection step turns out to be different.

2 Lagrange-Projection Decomposition and Finite-Volume Scheme

In this section, we briefly present the Lagrange-Projection decomposition considered in this paper and the corresponding first-order finite volume scheme.

Operator splitting decomposition and relaxation approximation. Using the chain rule for the space derivatives of (1), the Lagrange-Projection decomposition consists in first solving

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \partial _t \rho + \rho \partial _x u = 0, \\ \displaystyle \partial _t (\rho u) + \rho u \partial _x u + \partial _x p = 0, \\ \end{array} \right. \end{aligned}$$

(2)

which gives in Lagrangian coordinates $\tau \partial _x = \partial _m$, with $\tau =1/\rho$,

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \partial _t \tau - \partial _m u = 0, \\ \displaystyle \partial _t u + \partial _m p= 0, \\ \end{array} \right. \end{aligned}$$

(3)

and then the transport system

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \partial _t \rho + u \partial _x \rho = 0, \\ \displaystyle \partial _t (\rho u) + u \partial _x (\rho u) =0. \\ \end{array} \right. \end{aligned}$$

(4)

The numerical approximation of (3) and (4) will be given in the next sections but let us notice from now on that the Lagrangian system (3) will be treated considering the following relaxation approximation [6, 8],

$$\begin{aligned} \left\{ \begin{aligned}&\partial _{t}\tau - \partial _{m}u = 0, \\&\partial _{t}u + \partial _{m}\varPi = 0, \\&\partial _{t}\varPi + a^2\partial _{m}u = \lambda \left( p-\varPi \right) . \end{aligned} \right. \end{aligned}$$

(5)

Here, the new variable $\varPi$ represents a linearization of the real pressure p, the constant parameter a is a linearization of the Lagrangian sound speed $\rho c$ such that the sub-characteristic condition $a > \rho c$, $c=\sqrt{p'(\rho )}$, is satisfied, and the relaxation parameter $\lambda$ allows to recover $\varPi =p$ and the original system (3) in the asymptotic regime $\lambda \rightarrow \infty$. As usual, the relaxation system will be solved using a splitting strategy which consists in first setting $\varPi =p$ at initial time (which is formally equivalent to considering $\lambda \rightarrow \infty$ in (5)), and then solving the relaxation system (5) with $\lambda =0$.

First-order numerical scheme. The first-order finite volume scheme associated with the above decomposition and relaxation approximation is classical and given for instance in [2]. Nevertheless, it will be recovered in the DG extension proposed in the next section by setting the degree of all polynomials p to 0. Space and time will be discretized using a space step $\varDelta x$ and a time step $\varDelta t$. We will consider a set of cells $\kappa _j = [x_{j-1/2},x_{j+1/2})$ and instants $t^{n} = n \varDelta t$, where $x_{j+1/2}=j \varDelta x$ and $x_j = (x_{j-1/2} + x_{j+1/2})/2$ are respectively the cell interfaces and cell centers, for $j\in \mathbb {Z}$ and $n\in \mathbb {N}$.

3 Discontinuous Galerkin Discretization

We begin this section by introducing the notations of the DG discretization. Recall that the DG approach considers that the approximate solution at each time $t^n$ is defined on each cell $\kappa _j$ by a polynomial in space of order less or equal than p for a given integer $p\ge 1$ ($p=0$ corresponds to the usual first-order and piecewise constant finite volume scheme). With this in mind, we consider the $(p+1)$ Lagrange polynomials $\{\ell _i\}_{i=0,...,p}$ associated with the Gauss-Lobatto quadrature points in $[-1,1]$. More precisely, denoting $-1=s_0< s_1< \cdots < s_p = 1$ the $p+1$ Gauss-Lobatto quadrature points, $\ell _i$ is defined by the relations $\ell _i(s_k) = \delta _{i,k}$ for $k=0,...,p$, where $\delta$ is the Kronecker symbol. Then, in each cell $\kappa _j$, we define the shifted Lagrange polynomials $\varPhi _{i,j}$ by $\varPhi _{i,j}(x) = \ell _i \! \left( \frac{2}{\varDelta x}(x-x_j)\right)$ and we take $\{\varPhi _{i,j}\}_{i=0,...,p}$ as a basis for polynomials of order less or equal than p on $\kappa _j$. If we denote by $X_{\varDelta x}$ the DG approximation of X, we thus have $X_{\varDelta x}(x,t) = \sum _{k=0}^{p} X_{k,j}(t) \, \varPhi _{k,j}(x), \quad \forall x \in \kappa _j,$ where the coefficients $X_{k,j}$ depend on time and correspond to the value of the numerical solution at the shifted Gauss-Lobatto quadrature points $x_{k,j} = x_j + \frac{\varDelta x}{2} s_k$.

Before entering the details of the numerical approximation, let us briefly recall that the Gauss-Lobatto quadrature formula for $f : \kappa _j \times \mathbb {R}^+ \rightarrow \mathbb {R}$ writes

$$ \int _{\kappa _j} f(x,t) \,\mathrm {d}x \approx \frac{\varDelta x}{2} \sum _{k=0}^{p} \omega _k f(x_{k,j},t), $$

where $\omega _k$ are the weights of the Gauss-Lobatto quadrature. It is well-known that this formula is exact as soon as f is a polynomial of order less or equal than $(2p-1)$ with respect to x on $\kappa _j$. Just note that the integral $\int _{\kappa _j} \varPhi _{i,j}(x) \varPhi _{k,j}(x) \,\mathrm {d}x$ will be therefore approximated by $\frac{\varDelta x}{2} \omega _i \delta _{i,k}$ in the following. At last, note that the piecewise constant case $p=0$ can be also considered in this framework provided that we set $s_0 = 0$, $\varPhi _{0,j} = 1$ and $\omega _0 = 2$.

Time discretization $(t^n \rightarrow t^{n+1})$. We begin with the acoustic step (5) with $\lambda =0$. Multiplying the three equations by $\varPhi _{i,j}$, integrating over $\kappa _j$, and considering the piecewise polynomial approximations $X_{\varDelta x}$ for $X=\tau ,u, \varPi$ easily leads to

$$\begin{aligned} \left\{ \begin{aligned}&\frac{\varDelta x}{2} \omega _i \partial _t \tau _{i,j}(t) - \int _{\kappa _j} \varPhi _{i,j}(x)\partial _m u(x,t) \,\mathrm {d}x = 0, \\&\frac{\varDelta x}{2} \omega _i \partial _t u_{i,j}(t) + \int _{\kappa _j} \varPhi _{i,j}(x)\partial _m \varPi (x,t) \,\mathrm {d}x = 0, \\&\frac{\varDelta x}{2} \omega _i \partial _t \varPi _{i,j}(t) + a^2 \int _{\kappa _j} \varPhi _{i,j}(x)\partial _m u(x,t) \,\mathrm {d}x = 0, \end{aligned}\right. \end{aligned}$$

that we discretize in time by

$$\begin{aligned} \left\{ \begin{aligned} \tau _{i,j}^{n+1^-}&= \tau _{i,j}^{n} + \frac{2 \varDelta t}{\omega _i \varDelta x} \int _{\kappa _j} \varPhi _{i,j}(x)\partial _m u(x,t^{\alpha }) \,\mathrm {d}x, \\ u_{i,j}^{n+1^-}&= u_{i,j}^{n} - \frac{2 \varDelta t}{\omega _i \varDelta x}\int _{\kappa _j} \varPhi _{i,j}(x)\partial _m \varPi (x,t^{\alpha }) \,\mathrm {d}x, \\ \varPi _{i,j}^{n+1^-}&= \varPi _{i,j}^{n} - a^2\frac{2 \varDelta t}{\omega _i \varDelta x} \int _{\kappa _j} \varPhi _{i,j}(x)\partial _m u(x,t^{\alpha }) \,\mathrm {d}x, \end{aligned} \right. \end{aligned}$$

(6)

where the superscript $n+1^-$ formally represents the fictitious time $t^{n+1^-}$, and $\alpha = n$ or $\alpha = n+1^-$ if the time discretization is taken to be explicit or implicit.

As far as the transport step is concerned, the same process of reasoning leads to

$$\begin{aligned} \left\{ \begin{aligned} \rho _{i,j}^{n+1}&= \rho _{i,j}^{n+1^-} - \frac{2 \varDelta t}{\omega _i \varDelta x} \int _{\kappa _j} \varPhi _{i,j}(x) u(x,t^{\alpha }) \partial _x \rho (x,t^{n+1^-}) \,\mathrm {d}x, \\ (\rho u)_{i,j}^{n+1}&= (\rho u)_{i,j}^{n+1^-} - \frac{2 \varDelta t}{\omega _i \varDelta x}\int _{\kappa _j} \varPhi _{i,j}(x) u(x,t^{\alpha }) \partial _x (\rho u)(x,t^{n+1^-}) \,\mathrm {d}x. \end{aligned} \right. \end{aligned}$$

(7)

Note that this transport step is always treated explicitly in time.

Volume integrals and flux calculations. Considering the acoustic step, we aim at approximating the integrals $\int _{\kappa _j} \varPhi _{i,j}(x)\partial _m X(x,t^{\alpha }) \,\mathrm {d}x$ with $X=u, \varPi$. Observe that

$$\begin{aligned} \int _{\kappa _j} \varPhi _{i,j}(x)\partial _m X(x,t^{\alpha }) \,\mathrm {d}x&\approx \frac{\varDelta x}{2} \omega _i \tau _{i,j}^n \partial _x X(x_{i,j},t^{\alpha }) \,\mathrm {d}x = \tau _{i,j}^n \int _{\kappa _j} \varPhi _{i,j}(x)\partial _x X(x,t^{\alpha }) \,\mathrm {d}x, \end{aligned}$$

the last equality is indeed exact since X and $\varPhi$ are polynomials of order less or equal than p, so that $\varPhi _{i,j}\partial _x X(\cdot ,t)$ is of order less or equal than $(2p-1)$. The objective is now to use one integration by part to move the derivative from X to $\varPhi$, and to use the numerical fluxes to evaluate the interfacial terms, which gives

$$\begin{aligned} \int _{\kappa _j} \varPhi _{i,j}(x)\partial _x X(x,t^{\alpha }) \,\mathrm {d}x&\approx \delta _{i,p} X_{j+1/2}^{*,\alpha } - \delta _{i,0} X_{j-1/2}^{*,\alpha } - \frac{\varDelta x}{2} \sum _{k=0}^{p} \omega _k X_{k,j}^{\alpha } \partial _x \varPhi _{i,j}(x_{k,j}). \end{aligned}$$

Again, we refer the reader to [2] for the expressions of the star quantities in the above formula and the following ones, which are nothing but the numerical fluxes of the first-order finite volume scheme. At last, from (6) we obtain the acoustic step

$$\begin{aligned} \left\{ \begin{array}{rcl} \tau _{i,j}^{n+1^-} &{}=&{} \tau _{i,j}^{n} + \frac{2 \varDelta t}{\omega _i \varDelta x} \tau _{i,j}^n \,\Bigg [ \delta _{i,p} u_{j+1/2}^{*,\alpha } - \delta _{i,0} u_{j-1/2}^{*,\alpha } - \frac{\varDelta x}{2} \sum \nolimits _{k=0}^{p} \omega _k u_{k,j}^{\alpha } \partial _x \varPhi _{i,j}(x_{k,j})\Bigg ] \\ &{}=&{} L_{i,j}^{\alpha } \tau _{i,j}^n, \\ u_{i,j}^{n+1^-} &{}=&{} u_{i,j}^{n} - \frac{2 \varDelta t}{\omega _i \varDelta x} \tau _{i,j}^n \,\Bigg [ \delta _{i,p} \varPi _{j+1/2}^{*,\alpha } - \delta _{i,0} \varPi _{j-1/2}^{*,\alpha }- \frac{\varDelta x}{2} \sum \nolimits _{k=0}^{p} \omega _k \varPi _{k,j}^{\alpha } \partial _x \varPhi _{i,j}(x_{k,j})\Bigg ],\\ \varPi _{i,j}^{n+1^-} &{}=&{} \varPi _{i,j}^{n} - a^2\frac{2 \varDelta t}{\omega _i \varDelta x} \tau _{i,j}^n \,\Bigg [ \delta _{i,p} u_{j+1/2}^{*,\alpha } - \delta _{i,0} u_{j-1/2}^{*,\alpha }- \frac{\varDelta x}{2} \sum \nolimits _{k=0}^{p} \omega _k u_{k,j}^{\alpha } \partial _x \varPhi _{i,j}(x_{k,j})\Bigg ], \end{array} \right. \end{aligned}$$

(8)

with $L_{i,j}^{\alpha }= 1 + \frac{2 \varDelta t}{\omega _i \varDelta x} \,\Bigg [ \delta _{i,p} u_{j+1/2}^{*,\alpha } - \delta _{i,0} u_{j-1/2}^{*,\alpha } - \frac{\varDelta x}{2} \sum _{k=0}^{p} \omega _k u_{k,j}^{\alpha } \partial _x \varPhi _{i,j}(x_{k,j})\Bigg ].$

Regarding the transport step, we want to evaluate the integrals

$$ \int _{\kappa _j} \varPhi _{i,j}(x) u(x,t^{\alpha }) \partial _x X(x,t^{n+1^-}) \,\mathrm {d}x $$

with $X=\rho , \rho u$. The same process as before leads to

$$\begin{aligned}&\int _{\kappa _j} \varPhi _{i,j}(x) u(x,t^{\alpha }) \partial _x X(x,t^{n+1^-}) \,\mathrm {d}x = \delta _{i,p} X_{j+1/2}^{*,n+1^-} u_{j+1/2}^{*,\alpha } - \delta _{i,0} X_{j-1/2}^{*,n+1^-} u_{j-1/2}^{*,\alpha }\\&\qquad \quad \qquad \qquad \qquad - \int _{\kappa _j} (Xu) \partial _x \varPhi _{i,j} \,\mathrm {d}x - X_{i,j}^{n+1^-} \int _{\kappa _j} \varPhi _{i,j}(x) \partial _x u(x,t^{\alpha }) \,\mathrm {d}x, \end{aligned}$$

where we take

$\int _{\kappa _j} \varPhi _{i,j} \partial _x u(x,t^{\alpha }) \,\mathrm {d}x = \delta _{i,p} u_{j+1/2}^{*,\alpha } - \delta _{i,0} u_{j-1/2}^{*,\alpha } - \frac{\varDelta x}{2} \sum _{k=0}^{p} \omega _k u_{k,j}^{\alpha } \partial _x \varPhi _{i,j}(x_{k,j})$

and    $\int _{\kappa _j} (Xu) \partial _x \varPhi _{i,j} \,\mathrm {d}x \approx \frac{\varDelta x}{2} \sum _{k=0}^{p} \omega _k X_{k,j}^{n+1^-} u_{k,j}^{\alpha } \partial _x \varPhi _{i,j}(x_{k,j}).$

Conservativity property and mean values. Easy calculations not reported here show that the whole Lagrange-Projection scheme can be written as follows

${\displaystyle \rho _{i,j}^{n+1} = \rho _{i,j}^{n} - \frac{2 \varDelta t}{\omega _i \varDelta x} \,\left[ \delta _{i,p} \rho _{j+1/2}^{*,n+1^-} u_{j+1/2}^{*,\alpha } - \delta _{i,0} \rho _{j-1/2}^{*,n+1^-} u_{j-1/2}^{*,\alpha }- \frac{\varDelta x}{2} \sum _{k=0}^{p} \omega _k \rho _{k,j}^{n+1^-} u_{k,j}^{\alpha } \partial _x \varPhi _{i,j}(x_{k,j})\right] ,}$

${\displaystyle (\rho u)_{i,j}^{n+1} = (\rho u)_{i,j}^{n} - \frac{2 \varDelta t}{\omega _i \varDelta x} \,\left[ \delta _{i,p} \varPi _{j+1/2}^{*,\alpha } - \delta _{i,0} \varPi _{j-1/2}^{*,\alpha }- \frac{\varDelta x}{2} \sum _{k=0}^{p} \omega _k \varPi _{k,j}^{n+1^-} \partial _x \varPhi _{i,j}(x_{k,j})\right] }$

${\displaystyle - \frac{2 \varDelta t}{\omega _i \varDelta x} \,\left[ \delta _{i,p} (\rho u)_{j+1/2}^{*,n+1^-} u_{j+1/2}^{*,\alpha } - \delta _{i,0} (\rho u)_{j-1/2}^{*,n+1^-} u_{j-1/2}^{*,\alpha } - \frac{\varDelta x}{2} \sum _{k=0}^{p} \omega _k (\rho u)_{k,j}^{n+1^-} u_{k,j}^{\alpha } \partial _x \varPhi _{i,j}(x_{k,j})\right] }$

while the mean values $\overline{X}_{j}^{n+1} = \frac{1}{\varDelta x}\int _{\kappa _j} X(x,t^{n+1}) \,\mathrm {d}x = \sum _{i=0}^{p} \frac{\omega _i}{2} X_{i,j}^{n+1}$ with $X=\rho , \rho u$ obey the conservative formulas

$$\begin{aligned} \left\{ \begin{aligned} \overline{\rho }_{j}^{n+1}&= \overline{\rho }_{j}^{n} - \frac{\varDelta t}{\varDelta x} \left[ \rho _{j+1/2}^{*,n+1^-} u_{j+1/2}^{*,\alpha } - \rho _{j-1/2}^{*,n+1^-} u_{j-1/2}^{*,\alpha }\right] ,\\ \overline{(\rho u)}_{j}^{n+1}&= \overline{(\rho u)}_{j}^{n} - \frac{\varDelta t}{\varDelta x} \left[ \varPi _{j+1/2}^{*,\alpha } + (\rho u)_{j+1/2}^{*,n+1^-} u_{j+1/2}^{*,\alpha } \right. \\&\qquad \qquad \qquad \qquad \left. - \varPi _{j-1/2}^{*,\alpha } -(\rho u)_{j-1/2}^{*,n+1^-} u_{j-1/2}^{*,\alpha } \right] . \end{aligned} \right. \end{aligned}$$

(9)

Additional nonlinear stability properties can be proved for both the implicit and explicit schemes ($\alpha = n$ and $\alpha = n+1^-$). In particular, we have been able to prove the positivity of the nodal densities $\rho _{i,j}^{n+1^-}$ at time $t^{n+1^-}$ and of the mean densities $\overline{\rho }_{j}^{n+1}$ at time $t^{n+1}$, but also the validity of a discrete entropy inequality for the mean energy following the same lines as in [7].

Comparison with the double integration by part used in [7]. The present scheme turns out to be very close to the one recently proposed in [7], and it shares the same stability properties. However, the overall process in [7] is based on double integrations by part leading to the use of both numerical and exact fluxes at the interfaces, instead of only numerical fluxes in our approach. Interestingly, we observed that both schemes are strictly equivalent if one considers the mean values, but the nodal values turn out to be different because of the transport step. These little differences are due to the use of quadrature formulas to integrate the polynomials $X u \partial _x \varPhi _{i,j}$. In this case, the numerical integrations are not exact since polynomials $X u \partial _x \varPhi _{i,j}$ are of order $3p-1>2p-1$.

Positivity and generalized slope limiters. We have already stated the positivity of the nodal values $\rho _{i,j}^{n+1^-}$ at the end of the acoustic step and of the mean values $\overline{\rho }_{j}^{n+1}$ at the end of the transport step. Similarly to [7], we suggest to use a positivity limiter to ensure that $\rho _{i,j}^{n+1}>0$. More precisely, we replace $\rho _{i,j}^{n+1}$ by $\theta _{j} \rho _{i,j}^{n+1} + \left( 1-\theta _{j}\right) \overline{\rho }_{j}^{n+1},$ where the coefficients $\theta _{j}$ are taken to be $\theta _{j} = \min \left( 1,\frac{\overline{\rho }_j^{n+1} - \varepsilon }{\overline{\rho }_j^{n+1} - \min _i \rho _{i,j}^{n+1}}\right) .$ This formula ensures that if $\rho$ is less than the threshold $\varepsilon$, the nodal values of the corresponding cell are corrected, using the positive mean value, towards values greater than $\varepsilon$. In general we set the parameter $\varepsilon$ to $1.0\mathrm {e}^{-10}$. Note that in the forthcoming numerical experiments, the positivity limiter is not active. In order to avoid non physical oscillations, we also use the generalized slope limiters introduced in [4]. More precisely, considering the minmod function $m (a,b,c)=s \cdot \min (|a |,|b |,|c |)$ if

$s=\mathrm {sign}(a)=\mathrm {sign}(b)=\mathrm {sign}(c)$ and 0 otherwise, the increments

$\varDelta _+ \overline{X}_{j}^{n+1} = \overline{X}_{j+1}^{n+1} - \overline{X}_{j}^{n+1}$, $\varDelta _- \overline{X}_{j}^{n+1} = \overline{X}_{j}^{n+1} - \overline{X}_{j-1}^{n+1}$, and the values

         $X_{j+1/2}^{-,n+1} = \overline{X}_j^{n+1} + m\!\left( X_{p,j}^{n+1}-\overline{X}_j^{n+1}, \varDelta _+ \overline{X}_{j}^{n+1}, \varDelta _- \overline{X}_{j}^{n+1}\right)$,

         $X_{j-1/2}^{+,n+1} = \overline{X}_j^{n+1} - m\!\left( \overline{X}_j^{n+1}-X_{0,j}^{n+1}, \varDelta _+ \overline{X}_{j}^{n+1}, \varDelta _- \overline{X}_{j}^{n+1}\right)$,

the new states at time $t^{n+1}$ are defined by

$$\begin{aligned} {\left\{ \begin{array}{ll} X_{i,j}^{n+1} \qquad \qquad \qquad \qquad \qquad \text { if } X_{j+1/2}^{-,n+1} = X_{p,j}^{n+1} \text { and } X_{j-1/2}^{+,n+1} = X_{0,j}^{n+1},\\ \overline{X}_j^{n+1} + \frac{2}{\varDelta x} \left( {x_{i,j}}-x_j\right) \cdot m\!\left( \partial _x X^{n+1}(x_j),\varDelta _+ \overline{X}_{j}^{n+1}, \varDelta _- \overline{X}_{j}^{n+1}\right) \quad \text { otherwise.} \end{array}\right. } \end{aligned}$$

4 Numerical Results

The aim of this section is to compare our explicit-explicit $\text {EXEX}_p$ and implicit-explicit $\text {IMEX}_p$ Lagrange-Projection schemes, where p refers to the polynomial order of the DG approach. The time integrations are performed using Strong Stability Preserving Runge-Kutta methods described in [4]. Recall that $p(\rho )= g \rho ^2 /2$ so that the parameter a is chosen locally at each interface according to $a_{j+1/2} = \kappa \max \left( \rho _j^n \sqrt{g \rho _j^n} , \rho _{j+1}^n \sqrt{g \rho _{j+1}^n}\right)$ with $\kappa = 1.01$ and $g=9.81$. We set $\varDelta t=\min (\varDelta t_{\text {Lag}},\varDelta t_{\text {Tra}})$ for the $\text {EXEX}_p$ schemes and $\varDelta t=\varDelta t_{\text {Tra}}$ for the $\text {IMEX}_p$ schemes where $\varDelta t_{\text {Lag}} = \frac{\varDelta x}{2p+1}\min _j\left( 2 a_{j+1/2} \min (\tau _{p,j},\tau _{0,j+1})\right)$ is the DG time-step restriction associated with the Lagrangian step, while the Transport step CFL restriction reads $\varDelta t_{\text {Tra}} = \varDelta x \min _{i,j} \frac{2}{\omega _i} \left( \intop _{\kappa _j} u^{\alpha } \partial _x \varPhi _{i,j} \,\mathrm {d}x -\delta _p u_{j+1/2}^{*,\alpha ,-} + \delta _0 u_{j-1/2}^{*,\alpha ,+} \right) .$

Manufactured smooth solution. This preliminary test case is taken from [7] and allows us to test the experimental order of accuracy (EOA) of the schemes, especially on the Transport step. The space domain is $\left[ 0,1\right]$, the boundary conditions are periodic and the initial conditions are $\rho _0(x) = 1+0.2 \sin (2\pi x)$ and $u_0(x) = 1$. We solve (1) with a source term such that the exact solution is $\rho (x,t) = 1+0.2 \sin \left( 2\pi (x-t)\right)$ and $u(x,t) = 1$, which just means that we impose $u_{i,j}^{n+1^-}=1$ and $\varPi _{i,j}^{n+1^-}=\varPi _{i,j}^n$, so that the Acoustic step is trivial. Note that we use in this special case the $\text {EXEX}_p$ schemes. The EOA are reported in Table 1.

Dam break problem. In this test case, we take $\rho _0(x) = 20$ if $x \in \left[ 0,750\left[ \right. \right.$, $\rho _0(x) = 10$ if $x\in \left. \left. \right] 750,1500\right]$, and $u_0=0$ everywhere. The solutions given by the $\text {EXEX}_p$ and $\text {IMEX}_p$ schemes with $p=0$, 1 and 2 are shown on Fig. 1 using a 100-cell mesh, and compared with the classical first-order HLL scheme over a 100-cell mesh and a reference 1000-cell refined mesh. Note that the slope limiters allow to reduce spurious oscillations, but there is still a little undershoot for the $\text {EXEX}_1$ scheme.

Table 1 EOA for the manufactured smooth solution at time $T=0.5$

Fig. 1

Dam Break problem, water height at time $T=10$, $\text {EXEX}_p$ (left), $\text {IMEX}_p$ (right)