Finite Volume Multilevel Approximation of the Shallow Water Equations

Abstract

The authors consider a simple transport equation in one-dimensional space and the linearized shallow water equations in two-dimensional space, and describe and implement a multilevel finite-volume discretization in the context of the utilization of the incremental unknowns. The numerical stability of the method is proved in both cases.

Project supported by the National Science Foundation (Nos. DMS 0906440, DMS 1206438) and the Research Fund of Indiana University.

1 Introduction

This article is closely related to and complements the article (see [1]), in which the authors implemented multilevel finite-volume discretizations of the shallow water equations in two-dimensional space, as a model for geophysical flows. The geophysical context is presented in [1] as well as practical issues concerning the implementation. In this article, we recall the motivation, present the algorithm, and discuss the numerical analysis of some variations of the algorithm, and in particular the stability in time.

The shallow water equations are a simplified model of the primitive equations (or PEs for short) of the atmosphere and the oceans. As shown in [20, 24], in rectangular geometry, the PEs can be expanded by using a certain vertical modal decomposition. With such a decomposition, we obtain an infinite system of coupled equations, which resemble the shallow water equations. See [6, 7] for the actual numerical resolution of these coupled systems. However, it appears in these articles that the problems to be solved are very difficult (demanding), and performable numerical methods are needed to tackle more and more realistic problems. We turned to multilevel finite-volume methods in [1], finite-volume methods are desirable for the treatment of complicated geometrical domains such as the oceans, and multilevel methods of the incremental unknown type are useful for the implementation of multilevel methods. Such methods have been introduced in the context of the nonlinear Galerkin method in [18] (see also [19]), finite differences in [23], and spectral methods and turbulence in [8]. As continuation of [1], this article explores the finite-volume implementation of the incremental unknowns.

Considering to simplify a rectangular geometry, we divide the domain into cells of size Δx×Δy, which we regroup at the first level of increment, in cells of size 3Δx×3Δy. The unknowns on the small cells being the original unknowns, we introduce for the coarse cells suitably averaged values of the unknowns. The dynamic strategy, which may take many different forms (see [1, 8]), consists in solving alternatively the system for a number of time steps on the fine mesh grid and then for a number of time steps, the system considered on the coarse mesh during which the increments as defined below, remain frozen. This coarsening can be repeated once more considering cells of size 9Δx×9Δy, and possibly several times as the programming cost is repetitive and thus small, but we restrict ourselves in this article to one coarsening.

We have chosen to present the method for the shallow water (or SW for short) equations for the reasons mentioned above. We consider the SW equations without viscosity, linearized around a constant flow. The well-posedness of these linear hyperbolic equations has been established very recently (see [12]). We choose in this article one of many situations presented in [12], i.e., the fully supercritical case, since the boundary conditions depend on the nature of the flow (subcritical versus supercritical, subsonic versus supersonic). Other implementation of multilevel methods in geophysical fluid dynamics appear in [16]. See also [14, 15] for more developments on the primitive equations. Further developments along the lines of this work will appear in an article in [4].

Furthermore, some related results can be found in [2, 10, 11, 13, 17, 21, 25].

This article is organized as follows. We start in Sect. 2 with a simple model corresponding to a one-dimensional transport equation. We then proceed in Sect. 3 with the shallow water equation presenting first the equations (see Sect. 3.1), then the multilevel finite-volume discretization (see Sect. 3.2) and then the multilevel temporal discretization (see Sect. 3.3). In Sect. 4, we consider another related form of the algorithm. In Sects. 2 and 3, the algorithm on the coarse grid is the same as the algorithm on the fine grid (in space) with just a different spatial mesh. In this section, we consider another algorithm on which we started, where the spatial scheme on the coarse grid is obtained by averaging, in each coarse cell the equations for the corresponding fine cells. The study of the stability of the scheme in this case has not been completed yet. We present the analysis in one-dimensional space, for the simple transport equation (see Sect. 4.1) and for the one-dimensional linearized equation (see Sect. 4.2). The boundary condition is space periodicity and the stability analysis is conducted by the classical von Neumann method.

2 The One-Dimensional Case

We start with the one-dimensional space and consider the problem

$$\begin{aligned} \frac{\partial u}{\partial t}(x,t)+\frac{\partial u}{\partial x}(x,t)=f(x,t) \end{aligned}$$

(2.1)

for (x,t)∈(0,L)×(0,T), with the boundary condition

$$\begin{aligned} u(0,t)=0 \end{aligned}$$

(2.2)

and the initial condition

$$\begin{aligned} u(x,0)=u^0(x). \end{aligned}$$

(2.3)

We set $\mathcal{M}=(0,L)$ and $H=L^{2}(\mathcal{M})$, and also introduce the operator Au=u _x with domain $D(A)=\{ v\in H^{1}(\mathcal{M}),v(0)=0\}$. Then for f,f′∈L ¹(0,T;H), u ⁰∈D(A), problem (2.1)–(2.3) possesses a unique solution u, such that

$$u\in C\bigl([0,T];H\bigr)\cap L^\infty\bigl(0,T;D(A)\bigr),\quad\quad \frac{{\rm d}u}{{\rm d} t}\in L^\infty\bigl(0,T;D(A)\bigr). $$

Our multilevel spatial discretization is presented in Sect. 2.1, while Sect. 2.2 deals with time and space discretization.

2.1 Multilevel Spatial Discretization

We consider, on the interval (0,L), 3N cells (k _i )_1≤i≤3N of uniform length Δx with 3NΔx=L. For i=0,…,3N, we set

$$x_{i+\frac{1}{2}}=i\Delta x, $$

so that

$$k_i=(x_{i-\frac{1}{2}},x_{i+\frac{1}{2}}). $$

We also introduce the center of each cell,

$$x_i=\frac{x_{i-\frac{1}{2}}+x_{i+\frac{1}{2}}}{2}=(i-1)\Delta x+\frac {\Delta x }{2},\quad1\leq i \leq3N. $$

The discrete unknowns are denoted by u _i (1≤i≤3N), and u _i is expected to be some approximation of the mean value of u over k _i . Equation (2.1) integrated over the cell k _i yields

$$\frac{{\rm d}}{{\rm d}t}\int_{k_i}u(x,t){\rm d}x+u(x_{i+\frac{1}{2}},t)-u(x_{i-\frac{1}{2}},t)= \int_{k_i}f(x,t){\rm d}x. $$

Here the term $u(x_{i+\frac{1}{2}},t)$ is approximated by u _i (t) using an “upwind” scheme due to the direction of the characteristics for Eq. (2.1). Setting $f_{i}(t)=\frac{1}{\Delta x}\int_{k_{i}}f(x,t){\rm d}x$, the upwind finite-volume discretization now reads

$$\begin{aligned} \frac{{\rm d}u_i}{{\rm d}t}(t)+\frac {u_i(t)-u_{i-1}(t)}{\Delta x}=f_i(t), \quad1\leq i \leq3N, \end{aligned}$$

(2.4)

where we have set

$$\begin{aligned} u_0(t)=0. \end{aligned}$$

(2.5)

These equations are supplemented with the initial condition

$$\begin{aligned} u_i(0)=\frac{1}{\Delta x}\int_{k_i}u^0(x){ \rm d}x,\quad 1\leq i\leq3N. \end{aligned}$$

(2.6)

To rewrite the scheme in a more abstract form, we introduce the space V _h (h=Δx) of step functions u _h , which are constant on the intervals k _i , i=0,…,3N with $u_{h}|_{k_{i}}=u_{i}$ and u ₀=0. Here to take into account the boundary condition, we have added the fictitious cell k ₀=(−Δx,0). The discrete space V _h is equipped with the norm induced by $L^{2}(\mathcal{M})$, that is,

$$|u_h|^2=\Delta x\sum_{i=0}^{3N}|u_i|^2= \Delta x\sum_{i=1}^{3N}|u_i|^2. $$

Next let us introduce the backward difference operator

$$\partial_hu_h=\frac{u_i-u_{i-1}}{\Delta x}\quad{\rm on}~~k_i,\ 1\leq i \leq3N. $$

Then (2.4) can be rewritten as

$$\frac{{\rm d}u_h}{{\rm d}t}+\partial_hu_h=f_h $$

with $f_{h}|_{k_{i}}=f_{i}$.

We now introduce a coarser mesh consisting of the intervals K _l (1≤l≤N), with length 3Δx obtained as¹

$$\begin{aligned} K_l=k_{3l-2}\cup k_{3l-1} \cup k_{3l} =(x_{3l-2-\frac{1}{2}},x_{3l-\frac{1}{2}}). \end{aligned}$$

(2.7)

Let (u _i )_1≤i≤3N still denote the approximation of u on the fine mesh (k _i )_1≤i≤3N . Then an approximation of u on the coarse mesh is given by

$$\begin{aligned} U_{l}=\frac{1}{3}[ u_{3l-2}+u_{3l-1}+u_{3l}], \quad1\leq l \leq N. \end{aligned}$$

(2.8)

We introduce the incremental unknowns

$$\begin{aligned} Z_{3l-\alpha}=u_{3l-\alpha}-U_l \end{aligned}$$

(2.9)

for α=0,1,2, ℓ=1,…,N, so that

$$\begin{aligned} Z_{3\ell}+ Z_{3\ell-1} + Z_{3\ell-2} =0. \end{aligned}$$

(2.10)

Remark 2.1

The definition of Z in (2.9) is at our disposal. In this case, Z are the order of Δx. For example, using Taylor’s formula, we obtain

$$\begin{aligned} Z_{3l-2}& =u_{3l-2}-\frac{1}{3}[ u_{3l-2}+u_{3l-1}+u_{3l}] \\ &= {\frac{1}{3}\bigl[2u_{3l-2}-\bigl(u_{3l-2}+ \mathcal{O}(\Delta x)\bigr)-\bigl(u_{3l-2}+\mathcal{O}(\Delta x )\bigr) \bigr]} \\ &=\mathcal{O}(\Delta x). \end{aligned}$$

We will discuss elsewhere other definitions of the incremental unknown Z, and in particular those of order Δx ² considered in [1].

The unknowns on the fine grid are thus written as the sum of the coarse grid unknowns (U _l )_1≤l≤N and associated increments (Z _i )_1≤i≤3N .

With this in mind, we consider a coarse grid discretization of the equation similar to (2.4), that is,

$$\begin{aligned} \frac{{\rm d}U_{\ell}(t)}{{\rm d}t} + \frac{1}{3\Delta x} \bigl(U_\ell(t)-U_{\ell-1}(t) \bigr) = F_\ell(t),\quad1\leq\ell\leq N \end{aligned}$$

(2.11)

with

$$\begin{aligned} &U_0(t)=0, \end{aligned}$$

(2.12)

$$\begin{aligned} & F_\ell(t) = \frac{1}{3}\sum ^2_{\alpha=0}f_{3\ell-\alpha}(t) \end{aligned}$$

(2.13)

and

$$\begin{aligned} U_\ell(0) =\frac{1}{3}\sum ^2_{\alpha=0} u_{3\ell- \alpha}(0). \end{aligned}$$

(2.14)

Independent of the equation under consideration and the numerical scheme, let us make the following algebraic observation: for u _h ∈V _h , u _h =(u _i )_1≤i≤3N , we have

$$\begin{aligned} |u_h|^2 &=h\sum ^{3N}_{i=1} u^2_i = h\sum ^2_{\alpha=0}\sum^N_{\ell =1}|u_{3\ell-\alpha}|^2 \\ &=h\sum^2_{\alpha=0}\sum ^N_{\ell=1}|U_\ell+ Z_{3\ell -2}|^2 \\ & = 3h\sum^N_{\ell= 1} |U_\ell|^2 + h\sum^{3N}_{i=1} |Z_i|^2 \quad \bigl(\mbox{because of (2.10)} \bigr) \\ & = |U_h|^2 + |Z_h|^2 . \end{aligned}$$

(2.15)

In some sense, because of (2.10), the coarse component U and the increment Z are L ²-orthogonal.

2.2 Euler Implicit Time Discretization and Estimates

We define a time step Δt with N _T Δt=T, and set t _n =nΔt for 0≤n≤N _T . We denote by $\{ u_{i}^{n}, 1\leq i\leq3N,0\leq n\leq N_{T}\}$ the discrete unknowns. The value $u_{i}^{n}$ is an expected approximation

$$u_i^n\simeq\frac{1}{\Delta x}\int_{k_i}u(x,t_n){ \rm d}x. $$

Our spatial discretization was presented in the previous section in (2.4)–(2.6), for the fine grid, and (2.11)–(2.14) for the coarse grid. We now discretize this equation in time by using the implicit Euler scheme with the time step $\frac{\Delta t}{p}$ on the fine mesh and time step Δt on the coarse mesh. More precisely, let p>1 and q>1 be two fixed integers. The multi-step discretization consists in alternating p steps on (2.4) with time step $\frac{\Delta t}{p}$, from t _n to t _n+1 and then q steps on (2.11) with time step Δt, the incremental unknowns Z _i being frozen at t _n+1 from t _n+1 to t _n+q+1. Then, using equations (2.9), we can go back to the finer mesh for p steps from t _n+q+1 to t _n+q+2. For simplicity, we suppose that N _T is a multiple of q+1, and set $N_{q}=\frac{N_{T}}{q+1}$.

Suppose that n is a multiple of (q+1), and the $(u_{i}^{n})_{1\leq i \leq3N}$ are known. We introduce the discrete unknowns $u_{i}^{n+\frac{s}{p}}$ with $t_{n+\frac{s}{p}}=t_{n}+s\frac{\Delta t}{p}$ for 0≤s≤p and 1≤i≤3N. We successively determine the $u_{i}^{n+\frac{s}{p}}$ (1≤i≤3N, 1≤s≤p) with p iterations of the following scheme:

$$\begin{aligned} \left \{ \begin{array}{l} \frac{p}{\Delta t} (u_i^{n+\frac{s+1}{p}}-u_i^{n+\frac{s}{p}}) +\frac{1}{\Delta x} ( u_i^{n+\frac{s+1}{p}} - u_{i-1}^{n+\frac{s+1}{p}}) = f_i^{n+\frac {s+1}{p}},\\ {u_0^{n+\frac{s+1}{p}}=0}\\ \end{array} \right . \end{aligned}$$

(2.16)

for 1≤i≤3N, 0≤s≤p−1, where

$$\begin{aligned} f_i^{n+\frac{s+1}{p}}=\frac{1}{\frac{\Delta t}{p}} \frac{1}{\Delta x}\int_{(n+\frac{s}{p})\Delta t}^{(n+\frac{s+1}{p})\Delta t}\int _{k_i}f(x,t){\rm d} x{\rm d}t. \end{aligned}$$

(2.17)

It is convenient to introduce the step functions $u^{n+\frac{s}{p}}_{h}, \ f^{n+\frac{s}{p}}_{h}$ defined for 0≤s≤p by

$$u_h^{n+\frac{s}{p}}(x)=u_i^{n+\frac{s}{p}},\quad\quad f^{n+\frac{s}{p}}_h (x)=f^{n+\frac{s}{p}}_i,\quad x\in k_i,\ 1\leq i \leq3N. $$

We also introduce the backward difference operator ∂ _h defined by

$$\partial_hg^n_i = \frac{g^n_i-g^n_{i-1}}{\Delta x} \quad \text{or}\quad\partial_hg(x)=\frac{g(x)-g(x-h)}{\Delta x}, $$

so that (2.16) can now be rewritten as

$$\begin{aligned} \frac{p}{\Delta t} \bigl(u_h^{n+\frac{s+1}{p}} - u_h^{n+\frac{s}{p}}\bigr) + \partial_hu_h^{n+\frac{s+1}{p}} = f_h^{n+\frac{s+1}{p}}. \end{aligned}$$

(2.18)

Our goal now is to estimate $|u_{h}^{n+1}|$ in terms of $|u^{n}_{h}|$. We take the scalar product in $L^{2}(\mathcal{M})$ of (2.18) with $2\frac{\Delta t}{p} u_{h}^{n+\frac{s+1}{p}}$. Denoting by (⋅,⋅) the L ² scalar product and using the well-known relation

$$2(a-b,a)=|a|^2-|b|^2+|a-b|^2, $$

we find

$$\begin{aligned} & \big|u_h^{n+\frac{s+1}{p}}\big|^2 - \big|u^{n+\frac{s}{p}}_h\big|^2 +\big|u_h^{n+\frac{s+1}{p}}-u_h^{n+\frac{s}{p}}\big|^2+ \frac{2\Delta t}{p} \bigl(\partial_h u_h^{n+\frac{s+1}{p}}, u_h^{n+\frac{s+1}{p}}\bigr) \\ &\quad= \frac{2\Delta t}{p} \bigl(f_h^{n+\frac{s+1}{p}}, u_h^{n+\frac{s+1}{p}} \bigr). \end{aligned}$$

(2.19)

We have, for every u _h ∈V _h ,

$$\begin{aligned} 2(\partial_hu_h,u_h) =|u_{3N}|^2 +\sum^{3N}_{i=1} |u_i -u_{i-1}|^2. \end{aligned}$$

(2.20)

Indeed

$$\begin{aligned} 2(\partial_hu_h, u_h) & = 2\sum^{3N}_{i=1} (u_i-u_{i-1})u_i \\ & = \sum^{3N}_{i=1} \bigl(|u_i|^2-|u_{i-1}|^2 + |u_i-u_{i-1}|^2 \bigr), \end{aligned}$$

and (2.20) follows, since u ₀=0.

Using (2.20) and Schwarz inequality, (2.19) yields

$$\begin{aligned} & \big|u_h^{n+\frac{s+1}{p}}\big|^2 - \big|u^{n+\frac{s}{p}}_h\big|^2 + \big|u_h^{n+\frac {s+1}{p}}-u_h^{n+\frac{s}{p}}\big|^2 \\ &\quad\quad{}+ \frac{\Delta t}{p} \Biggl[ \big|u_{3N}^{n+\frac {s+1}{p}}\big|^2 + \sum^{3N}_{i=1} \big|u_i^{n+\frac{s+1}{p}} -u^{n+\frac {s+1}{p}}_{i-1}\big|^2 \Biggr] \\ &\quad\leq\frac{\Delta t}{p} \big|f_h^{n+\frac{s+1}{p}}\big|^2 + \frac {\Delta t}{p}\big|u_h^{n+\frac{s+1}{p}}\big|^2, \end{aligned}$$

(2.21)

so that

$$\begin{aligned} \biggl(1-\frac{\Delta t}{p} \biggr)\big|u_h^{n+\frac{s+1}{p}}\big|^2 \leq \frac{\Delta t}{p}\big|f_h^{n+\frac{s+1}{p}}\big|^2+\big|u_h^{n+\frac{s}{p}}\big|^2. \end{aligned}$$

(2.22)

This yields readily for 1≤s≤p,

$$\begin{aligned} \big|u_h^{n+\frac{s}{p}}\big|^2\leq \frac{1}{ (1-\frac{\Delta t}{p} )^s} \Biggl[ \big|u_h^n\big|^2+ \frac{\Delta t}{p} \sum_{d=0}^{s-1}\big|f_h^{n+\frac {d+1}{p}}\big|^2 \Biggr]. \end{aligned}$$

(2.23)

Here, in view of definition (2.17), we observe that

$$\begin{aligned} \frac{\Delta t}{p} \big|f_h^{n+\frac{d}{p}}\big|^2&= \frac{\Delta t}{p}\Delta x\sum_{i=1}^{3N}\big|f_i^{n+\frac{d}{p}}\big|^2= \biggl(\int_{(n+\frac{d}{p})\Delta t }^{(n+\frac{d+1}{p})\Delta t}\int^L_0f(x,t){ \rm d}x{\rm d}t \biggr)^2 \\ &\leq\int_{(n+\frac{d}{p})\Delta t}^{(n+\frac{d+1}{p})\Delta t}\int_0^L\big|f(x,t)\big|^2 {\rm d}x{\rm d} t. \end{aligned}$$

By adding these inequalities for d=0,…,p−1, we obtain

$$\frac{\Delta t}{p}\sum_{d=0}^{p-1} \big|f_h^{n+\frac{d}{p}} \big|^2\leq\int_{n\Delta t}^{(n+1)\Delta t} \int_0^L\big|f(x,t)\big|^2{\rm d}x{\rm d}t. $$

Combining this bound with (2.23) provides

$$\big|u_h^{n+\frac{s}{p}}\big|^2\leq\frac{1}{ (1-\frac{\Delta t}{p} )^s} \biggl[ \big|u_h^n\big|^2+\int_{n\Delta t}^{(n+1)\Delta t} \int_0^L\big|f(x,t)\big|^2{\rm d} x{\rm d}t \biggr]. $$

Since 1−x≥4^−x for $x\in [0,\frac{1}{2} ]$, we see that, if $\frac{\Delta t}{p}\leq\frac{1}{2}$,

$$\begin{aligned} \big|u_h^{n+\frac{s}{p}}\big|^2 \leq4^{\frac{s}{p}\Delta t} \biggl[ \big|u_h^n\big|^2+\int _{n\Delta t}^{(n+1)\Delta t}\int_0^L\big|f(x,t)\big|^2{ \rm d}x{\rm d}t \biggr]. \end{aligned}$$

(2.24)

Here s varies between 1 and p, and therefore the bound for s=p reads

$$\begin{aligned} \big|u_h^{n+1}\big|^2 \leq4^{\Delta t} \biggl[ \big|u_h^n\big|^2+\int _{n\Delta t}^{(n+1)\Delta t}\int_0^L\big|f(x,t)\big|^2{ \rm d}x{\rm d}t \biggr]. \end{aligned}$$

(2.25)

We now define the $u_{h}^{n+s}$ for 2≤s≤q+1, by applying q-times the implicit Euler scheme to Eq. (2.11) with step Δt, that is,

$$\begin{aligned} \left \{ \begin{array}{l} {\frac{U_l^{n+s+1}-U_l^{n+s}}{\Delta t}+\frac{U_{\ell }^{n+s+1}-U_{\ell-1}^{n+s+1}}{3\Delta x}=F_l^{n+s+1},}\\ {U_0^{n+s+1}=u_0^{n+s+1}=0,} \end{array} \right . \end{aligned}$$

(2.26)

where

$$\begin{aligned} F_l^{n+s+1}& =\frac {1}{3} \bigl[f_{3l-2}^{n+s+1}+f_{3l-1}^{n+s+1}+f_{3l}^{n+s+1} \bigr] \\ & =\frac{1}{3\Delta t\Delta x}\int_{(n+s)\Delta t}^{(n+s+1)\Delta t}\int _{K_l}f(x,t){\rm d} x {\rm d}t. \end{aligned}$$

(2.27)

As we said at the beginning of the section, the Z _i ’s are frozen between t _n+1 and t _n+q+1, and therefore for 2≤s≤q+1, 1≤l≤N,

$$\begin{aligned} \left \{ \begin{array}{l} {U_l^{n+s}=\frac {1}{3}[u_{3l-2}^{n+s}+u_{3l-1}^{n+s}+u_{3l}^{n+s}],}\\ Z_{3l-\alpha}^{n+s}=Z_{3l-\alpha}^{n+1}=u_{3l-\alpha }^{n+1}-U_l^{n+1},\quad\alpha=0,1,2.\\ \end{array} \right . \end{aligned}$$

(2.28)

We can invert this system (2.28) to obtain

$$\begin{aligned} u_{3l-\alpha}^{n+s}=U_l^{n+s}+Z_{3l-\alpha}^{n+1}, \quad\alpha= 0,1,2. \end{aligned}$$

(2.29)

Classically these equations allow us to uniquely define the terms $U^{n+s+1}_{\ell}$, when the terms $U^{n+1}_{\ell}$ are known. Then Eq. (2.29) allow us to compute the $u^{n+s+1}_{i}\ (i=1, \ldots, 3N,\ s=1,\ldots, q)$.

To derive suitable a priori estimates, we multiply (2.26) by $6\Delta t\Delta x U^{n+s+1}_{\ell}$ and sum for ℓ=1,…,N. Setting τ=n+s+1, we find

$$\begin{aligned} & 3\Delta x\sum^N_{\ell=1} \bigl(\big|U^\tau_\ell\big|^2 -\big|U^{\tau -1}_{\ell}\big|^2 \bigr)+3\Delta x\sum_{\ell=1}^N\big|U_l^\tau-U_l^{\tau -1}\big|^2 \\ &\quad\quad{}+ 2\Delta t\big|U^\tau_N\big|^2 +\Delta t \sum^N_{\ell= 1} \big|U^\tau _{\ell} - U^\tau_{\ell-1}\big|^2 \\ &\quad= 6\Delta t\Delta x\sum^N_{\ell= 1} F^\tau_\ell U^\tau_\ell. \end{aligned}$$

(2.30)

Hence, as for Eqs. (2.21)–(2.25),

$$\begin{aligned} \big|U^{\tau}_h\big|^2 \leq4^{\Delta t} \biggl[ \big|U^{\tau-1}_h\big|^2 +\int ^{\tau\Delta t}_{(\tau-1)\Delta t} \int^{L}_0 \big|f(x,t)\big|^2{\rm d}x{\rm d} t \biggr]. \end{aligned}$$

(2.31)

We write Eq. (2.31) for τ=n+2,…,n+q+1, multiply the equation for τ=n+s by 4^(q+1−s)Δt and add for s=2,…,q+1. We obtain

$$\begin{aligned} \begin{aligned} \big|U^{n+q+1}_h\big|^2 \leq4^{q\Delta t} \biggl[ \big|U^{n+1}_h\big|^2 + \int^{(n+q+1)\Delta t}_{(n+1)\Delta t}\int^L_0 \big|f(x,t)\big|^2 {\rm d}x{\rm d} t \biggr] . \end{aligned} \end{aligned}$$

(2.32)

We add $|Z^{n+1}_{h}|^{2}$ to both sides and, in view of (2.15) and the second formula of (2.28), we find

$$\begin{aligned} \big|u^{n+q+1}_h\big|^2 &\leq4^{q\Delta t} \biggl[ \big|u^{n+1}_h\big|^2 + \int^{(n+q+1)\Delta t}_{(n+1)\Delta t}\int^L_0\big| f(x,t)\big|^2{\rm d}x{\rm d} t \biggr] . \end{aligned}$$

(2.33)

Taking into account (2.25), we find that

$$\begin{aligned} \big|u^{n+q+1}_h\big|^2 \leq4^{(q+1)\Delta t} \biggl[\big|u^n_h\big|^2 + \int ^{(n+q+1)\Delta t}_{n\Delta t}\big|f(\cdot,t)\big|^2_2 {\rm d}t \biggr] . \end{aligned}$$

(2.34)

More generally, we have the stability result

$$\begin{aligned} \big|u^m_h\big|^2& \leq4^{m\Delta t} \biggl[ \big|u_h^0\big|^2 +\int ^{m\Delta t}_0 \big|f(\cdot,t)\big|^2_2 { \rm d}t \biggr] \\ &\leq4^T \biggl[ \big|u^0\big|^2 +\int ^T_0 \big|f(\cdot, t)\big|^2_{L^2}{ \rm d}t \biggr]. \end{aligned}$$

(2.35)

To summarize, we show the following result.

Theorem 2.1

The multilevel scheme defined by Eqs. (2.16) and (2.26) is stable in $L^{\infty}(0,T;L^{2}(\mathcal{M}))$ in the sense of (2.35).

3 The Linear Shallow Water Equations

We now want to extend the previous results to the more complex case of the shallow water equations linearized around a constant flow $(\tilde{u}_{0}, \tilde{v}_{0}, \tilde{\phi}_{0})$ (see (3.2) below). As shown in [12] the boundary conditions, which can be associated with these equations, depend on the relative values of the velocities ($\tilde{u}^{2}_{0},\tilde{v}^{2}_{0} >$ (or <) $g\tilde{\phi}_{0}$), that is, whether these velocities are sub- or supercritical (sub- or supersonic). We consider here the case, where

$$\begin{aligned} \tilde{\phi}_0>0,\quad\tilde{u}_0> \sqrt{g\tilde{\phi}_0}, \quad\tilde{v}_0>\sqrt{g \tilde{\phi}_0}. \end{aligned}$$

(3.1)

3.1 The Equations

We consider, in the domain $\mathcal{M}= (0,L_{1})\times(0,L_{2})$, the equations

$$\begin{aligned} \left \{ \begin{array}{l} {\frac{\partial u}{\partial t}+\tilde{u}_0\frac{\partial u}{\partial x}+\tilde{v}_0\frac {\partial u}{\partial y}+g\frac{\partial\phi}{\partial x}=f_u },\\ {\frac{\partial v}{\partial t}+\tilde{u}_0\frac{\partial v}{\partial x}+\tilde{v}_0\frac {\partial v}{\partial y}+g\frac{\partial\phi}{\partial y}=f_v },\\ {\frac{\partial\phi}{\partial t}+\tilde{u}_0\frac{\partial\phi }{\partial x}+\tilde{v}_0\frac{\partial\phi}{\partial y}+\tilde{\phi}_0 (\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y} )=f_\phi}. \end{array} \right . \end{aligned}$$

(3.2)

Here (u,v) is the velocity, and ϕ is the potential height. The advecting velocities $\tilde{u}_{0}$, $\tilde{v}_{0}$ and the mean geopotential height $\tilde{\phi}_{0}$ are constants. f=(f _u ,f _v ,f _ϕ ) is the source term. For the subcritical flow under consideration, we supplement (3.2) with the boundary conditions,

$$ \mathbf{u}=(u,v,\phi)=0, \quad\text{at }\{x=0\}\cup\{y=0\}, $$

(3.3)

and the initial conditions

$$ \mathbf{u} = (u,v,\phi) = \mathbf{u}^0 = \bigl(u^0,v^0,\phi^0\bigr),\quad \text{at } t= 0. $$

(3.4)

The system becomes

$$\frac{{\rm d}\mathbf{u}}{{\rm d}t}+\mathbf{A}\mathbf{u}=\mathbf{f}, $$

where Au=(A ₁ u,A ₂ u,A ₃ u) is given by

$$\begin{aligned} \begin{cases} {A_1\mathbf{u}=\tilde{u}_0\frac{\partial u}{\partial x}+\tilde {v}_0\frac{\partial u}{\partial y}+g\frac{\partial\phi}{\partial x}},\\ {A_2\mathbf{u}=\tilde{u}_0\frac{\partial v}{\partial x}+\tilde {v}_0\frac{\partial v}{\partial y}+g\frac{\partial\phi}{\partial y}},\\ {A_3\mathbf{u}=\tilde{u}_0\frac{\partial\phi}{\partial x}+\tilde {v}_0\frac{\partial\phi }{\partial y}+\tilde{\phi}_0 (\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y} )}. \end{cases} \end{aligned}$$

(3.5)

It may also be convenient to decompose A with respect to its x and y derivatives, that is,

$$\begin{aligned} \begin{aligned} &\mathbf{A} = \mathbf{A}^x + \mathbf{A}^y, \\ &\mathbf{A}^x\mathbf{u} = \bigl(A^x_1 \mathbf{u}, A^x_2\mathbf{u}, A^x_3 \mathbf{u}\bigr),\quad\mathbf{A}^y\mathbf{u} = \bigl(A^y_1 \mathbf{u}, A^y_2\mathbf{u}, A^y_3 \mathbf{u}\bigr) \end{aligned} \end{aligned}$$

with

$$\begin{aligned} \mathbf{A}^x\mathbf{u} = \begin{cases} \tilde{u}_0\frac{\partial u}{\partial x} + g\frac{\partial \phi }{\partial x},\\ \tilde{u}_0\frac{\partial v}{\partial x},\\ \tilde{u}_0\frac{\partial\phi}{\partial x} + \tilde{\phi }_0\frac{\partial u}{\partial x}, \end{cases} \quad\quad \mathbf{A}^y\mathbf{u} = \begin{cases} \tilde{v}_0\frac{\partial u}{\partial y},\\ \tilde{v}_0\frac{\partial v}{\partial y} + g \frac{\partial \phi }{\partial y},\\ \tilde{v}_0\frac{\partial\phi}{\partial y} +\tilde{\phi }_0\frac{\partial v}{\partial y}. \end{cases} \end{aligned}$$

We define the scalar product on $H=(L^{2}(\mathcal{M}))^{3}$ as follows: for u=(u,v,ϕ), u′=(u′,v′,ϕ′), and we set

$$\begin{aligned} \bigl\langle \mathbf{u},\mathbf{u}'\bigr\rangle = \bigl(u,u'\bigr) + \bigl( v,v'\bigr) + \frac {g}{\tilde{\phi}_0}\bigl( \phi,\phi'\bigr), \end{aligned}$$

(3.6)

where (⋅,⋅) denotes the standard scalar product on $L^{2}(\mathcal{M})$. Then the following positivity result for A holds.

Lemma 3.1

Under the assumption (3.1), for all sufficiently smooth u satisfying (3.3), we have 〈Au,u〉≥0.

Proof

We write

$$\begin{aligned} \langle\mathbf{A}\mathbf{u},\mathbf{u}\rangle= \bigl\langle \mathbf {A}^x\mathbf{u},\mathbf{u}\bigr\rangle +\bigl\langle \mathbf{A}^y\mathbf {u},\mathbf{u}\bigr\rangle \end{aligned}$$

(3.7)

with

$$\begin{aligned} \bigl\langle \mathbf{A}^x\mathbf{u},\mathbf{u}\bigr\rangle & = \iint_{\mathcal{M}} \biggl[ \tilde{u}_0u_xu+g \phi_xu+\tilde{u}_0 v_xv+ \frac{g}{\tilde{\phi}_0}\tilde{u}_0\phi_x\phi +gu_x\phi \biggr]{\rm d}x{\rm d}y, \\ \bigl\langle \mathbf{A}^y\mathbf{u},\mathbf{u}\bigr\rangle & = \iint_{\mathcal{M}} \biggl[ \tilde{v}_0v_yv+g \phi_yv+\tilde{v}_0 u_yu+ \frac{g}{\tilde{\phi}_0}\tilde{v}_0\phi_y\phi +gv_y\phi \biggr]{\rm d}x{\rm d}y. \end{aligned}$$

Then

$$\begin{aligned} \bigl\langle \mathbf{A}^x\mathbf{u},\mathbf{u}\bigr\rangle & =\frac {\tilde{u}_0 }{2}\iint_{\mathcal{M}} \biggl[\bigl(u^2 \bigr)_x+\bigl(v^2\bigr)_x+ \frac{g}{\tilde {\phi}_0}\bigl(\phi ^2\bigr)_x \biggr]{\rm d}x{ \rm d}y+\iint_{\mathcal{M}}g(\phi u)_x{\rm d}x{\rm d} y \\ & =\frac{\tilde{u}_0}{2}\int_0^{L_2} \biggl[u^2+v^2+\frac {g}{\tilde{\phi}_0}\phi^2 \biggr]_{x=0}^{x=L_1}{\rm d} y+\int_0^{L_2} \bigl[g(\phi u)\bigr]_{x=0}^{x=L_1}{\rm d}y. \end{aligned}$$

(3.8)

Recall that u=0 at x=0. Also the assumption (3.1) yields that

$$\begin{aligned} \frac{\tilde{u}_0}{2}u^2+\frac{\tilde{u}_0}{2}g\frac{\phi ^2}{\tilde{\phi}_0}+g\phi u \end{aligned}$$

is pointwise positive. Therefore, we infer from (3.8) that 〈A ^x u,u〉≥0. A similar computation provides 〈A ^y u,u〉≥0 (since $\tilde{v}_{0}^{2}>g\tilde{\phi}_{0}$). In view of (3.7), the proof of Lemma 3.1 is complete. □

Remark 3.1

The fact that the boundary and initial value problem (3.2)–(3.4) is well-posed is a recent result proved in [12]. The proof relies on the semigroup theory and necessitates in particular proving (by approximation) that 〈Au,u〉≥0 for all $\mathbf{u}\in L^{2}(\mathcal{M})^{3}$, such that $\mathbf{A}\mathbf{u}\in L^{2}(\mathcal{M})^{3}$, and u satisfies (3.3). The fact that (3.3) makes sense for such u’s results from a trace theorem also proved in [12].

3.2 Multilevel Finite-Volume Spatial Discretization

3.2.1 Finite-Volume Discretization

We decompose $\mathcal{M} = (0,L_{1})\times(0,L_{2})$ into 3N ₁×3N ₂ rectangles denoted by $(k_{i,j})_{1\leq i \leq 3N_{1},1\leq j \leq3N_{2}}$ of size Δx×Δy with 3N ₁Δx=L ₁ and 3N ₂Δy=L ₂.

For 0≤i≤3N ₁ and for 0≤j≤3N ₂, let

$$x_{i+\frac{1}{2}}=i\Delta x \quad{\rm and}\quad y_{j+\frac{1}{2}}=j\Delta y. $$

Then the rectangles (k _i,j ) are, for 1≤i≤3N ₁,1≤j≤3N ₂,

$$k_{i,j}=(x_{i-\frac{1}{2}},x_{i+\frac{1}{2}})\times(y_{j-\frac{1}{2}},y_{j+\frac{1}{2}}). $$

We also define the center (x _i ,y _j ) of each cell k _ij ,

$$\left \{ \begin{array}{l@{\quad}l} x_{i}=\frac{1}{2}(x_{i-\frac{1}{2}} + x_{i+\frac{1}{2}})=(i-1)\Delta x+\frac{\Delta x}{2}, & 1\leq i \leq 3N_1,\\ y_{j}=\frac{1}{2}(y_{j-\frac{1}{2}} + y_{j+\frac{1}{2}})=(j-1)\Delta y+\frac{\Delta y}{2}, & 1\leq j \leq3N_2.\\ \end{array} \right . $$

For the boundary conditions, we add fictitious cells on the west and south sides,

$$k_{0,j}=(-\Delta x,0)\times(y_{j-\frac{1}{2}},y_{j+\frac{1}{2}}),\quad \text{centered at } \biggl(x_0= -\frac{\Delta x}{2}, y_j \biggr), \quad 1\leq j\leq3N_2 $$

and

$$k_{i,0}=(x_{i-\frac{1}{2}}, x_{i+\frac{1}{2}})\times(-\Delta y,0), \quad \text{centered at } \biggl(x_i,y_0=- \frac{\Delta y}{2} \biggr),\quad 1\leq i\leq3N_1. $$

The finite-volume scheme is found by integrating the equations (3.2) over each control volume $(k_{i,j})_{1\leq i \leq 3N_{1},1\leq j \leq3N_{2}}$. The first equation yields for 1≤i≤3N ₁,1≤j≤3N ₂,

$$\begin{aligned} & \frac{{\rm d}}{{\rm d}t}\frac{1}{\Delta x\Delta y}\iint_{k_{i,j}}u(x,y,t){\rm d}x{\rm d}y +\frac{\tilde{u}_0}{\Delta x\Delta y}\int_{y_{j-\frac{1}{2}}}^{y_{j+\frac{1}{2}}} \bigl[u(x_{i+\frac{1}{2}},y,t)-u(x_{i-\frac{1}{2}},y,t)\bigr]{\rm d}y \\ &\quad{}+\frac{\tilde{v}_0}{\Delta x\Delta y}\int_{x_{i-\frac{1}{2}}}^{x_{i+\frac{1}{2}}} \bigl[u(x,y_{j+\frac{1}{2}},t)-u(x,y_{j-\frac{1}{2}},t)\bigr]{\rm d}x \\ &\quad{}+\frac{g}{\Delta x\Delta y}\int_{y_{j-\frac{1}{2}}}^{y_{j+\frac{1}{2}}}\bigl[ \phi(x_{i+\frac{1}{2}},y,t)-\phi(x_{i-\frac{1}{2}},y,t)\bigr]{\rm d} y=\int _{k_{i,j}}f_u(x,y,t){\rm d}x{\rm d}y. \end{aligned}$$

Let us denote

$$\begin{aligned} V_h =&\{\text{the space of step functions constant on } k_{i,j},\ 0\leq i \leq3N_1, 0\leq j \leq3N_2\ \text{with} \\ &\phantom{\{} w_{|k_{i,j}}=w_{i,j}\mbox{ and } w_{0,j}=w_{i,0}=0\}. \end{aligned}$$

We approximate the unknown u=(u,v,ϕ) with u _h ≃u _h (t)∈(V _h )³=V _h , and use an upwind scheme for the fluxes, since $\tilde{u}_{0}>0$ and $\tilde{v}_{0}>0$,

$$\begin{aligned} & \mathbf{u}(x_{i+\frac{1}{2}},y,t)\simeq\mathbf{u}_{i,j}(t),\quad y \in [y_{j-\frac{1}{2}},y_{j+\frac{1}{2}}], \\ & \mathbf{u}(x,y_{j+\frac{1}{2}},t)\simeq\mathbf{u}_{i,j}(t),\quad x \in [x_{i-\frac{1}{2}},x_{i+\frac{1}{2}}]. \end{aligned}$$

This gives the following semi-discrete equations for 1≤i≤3N ₁ and 1≤j≤3N ₂:

$$\begin{aligned} \left \{ \begin{array}{l} {\frac{{\rm d}}{{\rm d}t}u_{i,j}+\tilde{u}_0\frac {u_{i,j}-u_{i-1,j}}{\Delta x}+\tilde{v}_0\frac{u_{i,j}-u_{i,j-1}}{\Delta y}+g\frac{\phi_{i,j}-\phi_{i-1,j}}{\Delta x}=f_{u,i,j},}\\ {\frac{{\rm d}}{{\rm d}t}v_{i,j}+\tilde{u}_0\frac {v_{i,j}-v_{i-1,j}}{\Delta x}+\tilde{v}_0\frac{v_{i,j}-v_{i,j-1}}{\Delta y} +g\frac{\phi_{i,j}-\phi_{i,j-1}}{\Delta y}=f_{v,i,j},}\\ {\frac{{\rm d}}{{\rm d}t}\phi_{i,j}+\tilde{u}_0\frac{\phi _{i,j}-\phi _{i-1,j}}{\Delta x}+\tilde{v}_0\frac{\phi_{i,j}-\phi _{i,j-1}}{\Delta y}}\\ \quad{}+\tilde{\phi}_0 (\frac{u_{i,j}-u_{i-1,j}}{\Delta x} +\frac{v_{i,j}-v_{i,j-1}}{\Delta y} )=f_{\phi,i,j},\\ {\mathbf{u}_{0,j}=\mathbf{u}_{i,0}=0},\\ {\mathbf{u}_{i,j}(0)=\mathbf{u}_{i,j}^0},\\ \end{array} \right . \end{aligned}$$

(3.9)

where f=(f _u ,f _v ,f _ϕ ), $\mathbf{u^{0}}=(u^{0},v^{0},\phi ^{0})$ and

$$\begin{aligned} \mathbf{f}_{i,j}(t)=\frac{1}{\Delta x\Delta y}\int_{k_{i,j}} \mathbf {f}(x,y,t){\rm d}x{\rm d}y,\quad\quad{\mathbf{u}^0_{i,j}}= \frac{1}{\Delta x\Delta y }\int_{k_{i,j}}\mathbf{u}^0(x,y){\rm d}x{\rm d}y. \end{aligned}$$

(3.10)

Let us introduce the finite difference operators

$$\begin{aligned} & \partial_{1h}g_h=\frac{1}{\Delta x}(g_{i,j}-g_{i-1,j}) \quad{\rm on}\ k_{i,j}, \\ & \partial_{2h}g_h=\frac{1}{\Delta y}(g_{i,j}-g_{i,j-1}) \quad{\rm on}\ k_{i,j}. \end{aligned}$$

We can now define in an obvious way, based on (3.9) the finite difference operator A _h =(A _1h ,A _2h ,A _3h ), operating on V _h

$$\begin{aligned} \begin{cases} A_{1h}\mathbf{u}_h=\tilde{u}_0\partial_{1h}u_h+\tilde {v}_0\partial_{2h}u_h+g\partial _{1h}\phi_h,\\ A_{2h}\mathbf{u}_h=\tilde{u}_0\partial_{1h}v_h+\tilde {v}_0\partial_{2h}v_h+g\partial _{2h}\phi_h,\\ A_{3h}\mathbf{u}_h=\tilde{u}_0\partial_{1h}\phi_h+\tilde {v}_0\partial_{2h}\phi_h+\tilde{\phi}_0 \partial_{1h}u_h+\tilde{\phi}_0\partial_{2h}v_h\\ \end{cases} \end{aligned}$$

(3.11)

and its decomposition $\mathbf{A}_{h}=\mathbf{A}^{x}_{h} + \mathbf{A}^{y}_{h}$, to be used later on,

$$\begin{aligned} \begin{cases} A^x_{h}\mathbf{u}_h=(\tilde{u}_0\partial_{1h}u_h+g\partial _{1h}\phi _h,\tilde{u}_0\partial _{1h}v_h ,\tilde{u}_0\partial_{1h}\phi_h+\tilde{\phi }_0\partial_{1h}u_h ),\\ A^x_{h}\mathbf{u}_h=(\tilde{v}_0\partial_{2h}u_h,\tilde {v}_0\partial_{2h}v_h+g\partial _{2h}\phi_h,\tilde{v}_0\partial_{2h}\phi_h+\tilde{\phi }_0\partial_{2h}v_h). \end{cases} \end{aligned}$$

(3.12)

Those are the discrete versions of A,A ₁,A ₂,A ₃,A ^x,A ^y.

We can now check that A _h , the discrete version of A, is positive like A.

Lemma 3.2

For all u _h =(u _h ,v _h ,ϕ _h )∈V _h , we have

$$\begin{aligned} \langle\mathbf{A}_h \mathbf{u}_h, \mathbf{u}_h \rangle\geq0, \end{aligned}$$

(3.13)

where 〈⋅,⋅〉 is the scalar product on $L^{2}(\mathcal{M})^{3}$, given by (3.6).

Proof

We write

$$\begin{aligned} \langle\mathbf{A}_h\mathbf{u}_h,\mathbf{u}_h\rangle=\langle A^x_h\mathbf{u}_h,\mathbf{u}_h\rangle+\langle A^y_h\mathbf {u}_h,\mathbf{u}_h\rangle, \end{aligned}$$

(3.14)

where

$$\begin{aligned} \bigl\langle A^x_h\mathbf{u}_h, \mathbf{u}_h\bigr\rangle & = {(\tilde {u}_0\partial _{1h}u_h,u_h)+(g\partial_{1h} \phi_h,u_h)+(\tilde{u}_0\partial _{1h}v_h,v_h)} \\ &\quad {}+ \frac{g}{\tilde{\phi}_0}(\tilde{u}_0\partial _{1h} \phi_h,\phi_h)+g(\partial _{1h}u_h, \phi_h ), \\ \bigl\langle A^y_h\mathbf{u}_h, \mathbf{u}_h\bigr\rangle & = (\tilde {v}_0\partial _{2h}u_h,u_h)+(g\partial_{2h} \phi_h,v_h)+(\tilde{v}_0\partial _{2h}v_h,v_h) \\ & \quad{}+ \frac{g}{\tilde{\phi}_0}(\tilde{v}_0\partial_{2h} \phi _h,\phi_h)+g(\partial_{2h}v_h, \phi_h ). \end{aligned}$$

We first remark that

$$\begin{aligned} (\tilde{u}_0\partial_{1h}u_h,u_h)&= \frac{\tilde {u}_0}{2}\Delta y\sum_{i=1}^{3N_1} \sum_{j=1}^{3N_2}\bigl(|u_{i,j}|^2-|u_{i-1,j}|^2+|u_{i,j}-u_{i-1,j}|^2 \bigr) \\ &=\frac{\tilde{u}_0}{2}\Delta y\sum_{j=1}^{3N_2} \Biggl(|u_{3N_{1,j}}|^2+\sum_{i=1}^{3N_1}|u_{i,j}-u_{i-1,j}|^2 \Biggr). \end{aligned}$$

(3.15)

Then we write

$$\begin{aligned} & (\phi_{i,j}-\phi_{i-1,j})u_{i,j}+(u_{i,j}-u_{i-1,j}) \phi _{i,j}\\ &\quad=u_{i,j}\phi _{i,j}-u_{i-1,j} \phi_{i-1,j}+(u_{i,j}-u_{i-1,j}) (\phi_{i,j}- \phi_{i-1,j}). \end{aligned}$$

Using these two formulas, we obtain

$$\begin{aligned} & (\tilde{u}_0\partial_{1h}u_h,u_h)+ \frac{g}{\tilde {\phi}_0}(\tilde{u}_0\partial_{1h}\phi _h,\phi_h)+(g\partial_{1h} \phi_h,u_h)+g(\partial_{1h}u_h, \phi_h ) \\ &\quad=\frac{\tilde{u}_0}{2}\Delta y\sum_{j} \biggl(|u_{3N_1,j}|^2+\frac{g}{\tilde{\phi}_0 }|\phi_{3N_1,j}|^2 \biggr) \\ &\quad\quad{}+\Delta y\frac{\tilde{u}_0}{2}\sum_{i,j}|u_{i,j}-u_{i-1,j}|^2+ \Delta y\frac{g\tilde{u}_0}{2\tilde{\phi}_0}\sum_{i,j}|\phi _{i,j}-\phi_{i-1,j}|^2 \\ &\quad\quad{}+g\Delta y\sum_{i,j}(u_{i,j}-u_{i-1,j}) (\phi_{i,j}-\phi_{i-1,j})+g\Delta y\sum _{j}u_{3N_1,j}\phi_{3N_1,j}. \end{aligned}$$

Since $\tilde{u}_{0}>0$ and $\tilde{u}_{0}^{2}>g\tilde{\phi}_{0}$, the expressions

$$\frac{\tilde{u}_0}{2}|u_{i,j}-u_{i-1,j}|^2+ \frac{g\tilde {u}_0}{2\tilde{\phi}_0}|\phi_{i,j}-\phi _{i-1,j}|^2+g(u_{i,j}-u_{i-1,j}) (\phi_{i,j}-u\phi_{i-1,j}) $$

and

$$\frac{\tilde{u}_0}{2}|u_{3N_1,j}|^2+\frac{g\tilde {u}_0}{2\tilde{\phi}_0}|\phi _{3N_1,j}|^2+gu_{3N_1,j}\phi_{3N_1,j} $$

are positive and the corresponding sums are positive as well.

Finally, using also the analogue of (3.15) for v _h , we conclude that $\langle A^{x}_{h}\mathbf{u}_{h},\mathbf{u}_{h}\rangle\geq 0$. Similarly, it can be checked that $\langle A^{y}_{h}\mathbf{u}_{h},\mathbf{u}_{h}\rangle\geq0$. Recalling (3.14), this completes the proof of Lemma 3.2. □

In fact, a perusal of the calculations above shows that we have proved the following useful lemma.

Lemma 3.3

For every u _h ∈V _h ,

$$\begin{aligned} \begin{cases} \langle\mathbf{A}^x_h\mathbf{u}_h,\mathbf{u}_h\rangle\geq \kappa_1\Delta y\sum^{3N_2}_{j=1} \bigl[|\mathbf{u}_{3N1,j}|^2+\sum^{3N_{1}}_{i=1}|\mathbf{u}_{i,j}-\mathbf{u}_{i-1,j}|^2 \bigr],\\ \langle\mathbf{A}^y_h, \mathbf{u}_h,\mathbf{u}_h\rangle\geq \kappa_1\Delta x\sum^{3N_1}_{i=1} \bigl[|\mathbf{u}_{i,3N_{2}}|^2 +\sum^{3N_{1}}_{j=1}|\mathbf{u}_{i,j} - \mathbf{u}_{i,j-1}|^2 \bigr], \end{cases} \end{aligned}$$

(3.16)

where the constant κ ₁ depends on $\tilde{u}_{0},\tilde{v}_{0},\tilde{\phi}_{0},g$ and in particular on the positive numbers $\tilde{u}^{2}_{0}-g\tilde{\phi}_{0}, \tilde{v}^{2}_{0} - g\tilde{\phi}_{0}$.

3.2.2 Multilevel Finite-Volume Discretization

We introduce the coarse mesh consisting of the rectangles K _lm (1≤l≤N ₁,1≤m≤N ₂),²

$$\begin{aligned} K_{lm} = \bigcup^2_{\alpha,\beta=0} k_{3l-\alpha, 3m-\beta} = ( x_{3l-2-\frac{1}{2}}, x_{3l-\frac{1}{2}}) \times(y_{3m-2-\frac{1}{2}}, y_{3m+\frac{1}{2}}). \end{aligned}$$

We also define the fictitious rectangles K _0,m ,K _l,0 (l=1,…N ₁, m=1,…,N ₂), needed for the implementation of the boundary conditions, and they are defined as above with m or l=0.

We introduce the space V _3h defined like V _h . If u _h ∈V _h and $u_{h}|_{k_{ij}} = u_{i,j}$, we define for l=1,…,N ₁, m=1,…,N ₂ the averages as

$$\begin{aligned} U_{l,m} = \frac{1}{9}\sum ^2_{\alpha,\beta=0} u_{3l-\alpha,3m-\beta}, \end{aligned}$$

(3.17)

and the incremental unknowns as

$$\begin{aligned} Z_{3l-\alpha,3m-\beta} = u_{3l-\alpha,3m-\beta}-U_{l,m}, \end{aligned}$$

(3.18)

which satisfy of course

$$\begin{aligned} \sum^2_{\alpha,\beta=0} Z_{3l-\alpha,3m-\beta}=0. \end{aligned}$$

(3.19)

We note the following algebraic relations (using (3.19)):

$$\begin{aligned} \sum^2_{\alpha,\beta=0} |u_{3l-\alpha,3m-\beta}|^2=9|U_{l,m}|^2 + \sum ^2_{\alpha,\beta=0}|Z_{3l-\alpha,3m-\beta}|^2. \end{aligned}$$

(3.20)

Multiplying by ΔxΔy and adding for l=1,…,N ₁, m=1,…,N ₂, we find

$$\begin{aligned} |u_h|^2 =|U_h|^2 +|Z_h|^2, \end{aligned}$$

(3.21)

where |⋅| is still the norm in $L^{2}(\mathcal{M}), U_{h}$ is the step function equal to U _lm on K _l,m and Z _h is the step function equal to Z _i,j on k _i,j .

3.3 Euler Implicit Time Discretization and Estimates

We proceed to some extent as in the one-dimensional space. We define a time step Δt with N _T Δt=T, and set t _n =nΔt. We denote by

$$\begin{aligned} \mathbf{u}^n_h = \bigl\{ \mathbf{u}^n_{i,j}, \ 1\leq i\leq3N_1,\ 1\leq j\leq3N_2\bigr\} \end{aligned}$$

the discrete unknowns, where $\mathbf{u}^{n}_{i,j}$ is an expected approximation

$$\mathbf{u}^n_{i,j}\simeq\frac{1}{\Delta x\Delta y}\int _{k_{i,j}}\mathbf{u} (x,y,t_n){\rm d}x{\rm d}y. $$

The spatial discretization has been presented in Sect. 3.2. We will now discretize the shallow water equations in time by using the implicit Euler scheme, and advance equation (3.9) for p steps in time on the fine mesh with a time step of $\frac{\Delta t}{p}$, where p (and q below) are two fixed integers larger than 1.

These steps will bring us, e.g., from t _n to t _n+1. We then perform q steps with a time step Δt bringing us from t _n+1 to t _n+q+1. For simplicity, we suppose that N _T is a multiple of q+1, and we set $N_{q}=\frac{N_{T}}{q+1}$. The steps performed with the time step Δt will use the coarse mesh. We first consider in Sect. 3.3.1 the p steps performed with mesh $\frac{\Delta t}{p}$ on the fine grid. Then the q steps on the coarse grid are described in Sect. 3.3.2.

3.3.1 Scheme and Estimates on the Fine Grid

We start from Eqs. (3.9) and write thus for s=1,…,p,

$$\begin{aligned} \begin{cases} \frac{p}{\Delta t}\bigl(u^{n+\frac{s+1}{p}}_{i,j} - u^{n+\frac {s}{p}}_{i,j}\bigr)+ \tilde{u}_0\partial_{1h}u_{i,j}^{n+\frac{s+1}{p}} \\ \quad{}+\tilde{v}_0\partial_{2h}u^{n+\frac{s+1}{p}}_{i,j} + g\partial_{1h}\phi^{n+{\frac{s+1}{p}}}_{i,j} = f_{u,i,j}^{n+\frac{s+1}{p}},\\ \frac{p}{\Delta t}\bigl(v^{n+\frac{s+1}{p}}_{i,j} - v^{n+\frac{s}{p}}_{i,j}\bigr) + \tilde{u}_0\partial_{1h}v_{i,j}^{n+\frac{s+1}{p}}\\ \quad{}+\tilde{v}_0\partial_{2h}v^{n+(s+\frac{1}{p})}_{i,j} + g\partial_{2h}\phi^{n+\frac{s+1}{p}}_{i,j} = f^{n+\frac{s+1}{p}}_{v,i,j},\\ \frac{p}{\Delta t}\bigl(\phi^{n+\frac{s+1}{p}}_{i,j} - \phi^{n+\frac {s}{p}}_{i,j}\bigr) + \tilde{u}_0\partial_{1h}\phi^{n+\frac{s+1}{p}}_{i,j}\\ \quad{}+\tilde{v}_0\partial_{2h}\phi^{n+\frac{s+1}{p}}_{i,j} + \tilde{\phi}_0\bigl(\partial_{1h}u_{i,j}^{n+\frac{s+1}{p}} +\partial_{2h}v^{n+\frac{s+1}{p}}_{i,j}\bigr) = f^{n+\frac{s+1}{p}}_{\phi,i,j}. \end{cases} \end{aligned}$$

(3.22)

With the definition of A _h introduced in (3.11), Eq. (3.22) amount to

$$\begin{aligned} \frac{p}{\Delta t}\bigl(\mathbf{u}^\tau_h - \mathbf{u}^{\tau-\frac{1}{p}}_h\bigr) + A_h \mathbf{u}^\tau_h=\mathbf{f}_h^\tau. \end{aligned}$$

(3.23)

Here we have set for simplicity $n+\frac{s+1}{p}=\tau,\ n+\frac{s}{p} =\tau-\frac{1}{p},\ \mathbf{u}^{\tau}_{h} = (u^{\tau}_{h}, v^{\tau}_{h}, \phi^{\tau}_{h}), \mathbf{f}^{\tau}_{h} = (f_{u,h}^{\tau}, f_{v,h}^{\tau}, f_{\phi, h}^{\tau})$.

Taking the scalar product in V _h of each side of (3.23) with $2\frac{\Delta t}{p}\mathbf{u}^{\tau}$, we see that

$$\begin{aligned} & \big|\mathbf{u}^\tau_h\big|^2-\big| \mathbf{u}^{\tau-\frac{1}{p}}_h\big|^2 +\big|\mathbf{u}^\tau_h - \mathbf{u}^{\tau-\frac{1}{p}}_h\big|^2+2\frac {\Delta t}{p} \bigl\langle \mathbf{A}_h\mathbf{u}_h^\tau, \mathbf {u}_h^\tau\bigr\rangle \\ &\quad=\frac{2\Delta t}{p} \bigl\langle \mathbf{f}_h ^\tau, \mathbf{u}^\tau_h\bigr\rangle \leq\frac{\Delta t}{p}\big| \mathbf{f}_h ^\tau\big|^2 + \frac{\Delta t}{p}\big| \mathbf{u}_h ^\tau\big|^2. \end{aligned}$$

(3.24)

Hence thanks to Lemma 3.2 (comparing with (2.19)–(2.25)),

$$\begin{aligned} \big|\mathbf{u}_h^{n+\frac{s+1}{p}}\big|^2 \leq\frac{1}{1-\frac{\Delta t}{p}} \big|\mathbf{u}_h^{n+\frac{s}{p}}\big|^2 + \frac{1}{1-\frac{\Delta t}{p}}\frac{\Delta t}{p}\big|\mathbf{f}_h^{n+\frac{s}{p}}\big|^2, \end{aligned}$$

(3.25)

and for $\frac{\Delta t}{p}\leq\frac{1}{2}$ and s=1,…,p (comparing with (2.25)),

$$\begin{aligned} \begin{aligned} \big|\mathbf{u}_h^{n+\frac{s}{p}}\big|^2& \leq4^{\frac{s\Delta t}{p} }\kappa ^n\bigl( \mathbf{u}^0,\mathbf{ f }\bigr), \\ \kappa^n\bigl(\mathbf{u}^0,\mathbf{f}\bigr)& = \big| \mathbf{u}^0_{h}\big|^2+\int_{n\Delta t} ^{(n+1)\Delta t}\int_0^{L_2}\int _0^{L_1}\big|\mathbf{ f }(x,y,t)\big|^2{\rm d} x{\rm d} y{\rm d}t. \end{aligned} \end{aligned}$$

In particular, for s=p,

$$\begin{aligned} \big|\mathbf{u}_h^{n+1}\big|^2 \leq4^{\Delta t}\kappa^n\bigl(\mathbf{u}^0,\mathbf{f} \bigr). \end{aligned}$$

(3.26)

3.3.2 Scheme and Estimates on the Coarse Grid

We now consider the q a time-steps performed on the coarse grid with a time step Δt.

We discretize Eq. (3.9) in time, starting from time t _n+1=(n+1)Δt using the same scheme as for Eq. (3.22) but with a coarse mesh (comparing with (2.26)). We obtain

$$\begin{aligned} \frac{1}{\Delta t}\bigl(\mathbf{U}^\tau_h- \mathbf{U}^{\tau-1}_{h}\bigr) +\mathbf{A}_{3h} \mathbf{U}^\tau_h = \mathbf{F}^\tau_h, \end{aligned}$$

(3.27)

where $\tau=n+s +1,\ s =1,\ldots, q,\ \mathbf{U}^{\tau}_{h} =(U^{\tau}_{u,h}, U^{\tau}_{v,h}, U^{\tau}_{\phi,h})$ and U _h ∈V _3h has components U _i,j on K _i,j (i=0,…,N ₁, j=0,…,N ₂). Finally, $\mathbf{F}^{\tau}_{h}$ has components $\mathbf{F}^{\tau}_{i,j}$ on K _i,j with

$$\begin{aligned} \mathbf{F}^\tau_{i,j} =\frac{1}{\Delta t} \frac{1}{9\Delta x\Delta y}\int^{\tau\Delta t}_{(\tau-1)\Delta t}\int _{K_{i,j}}\mathbf{f} (x,y,t){\rm d} x{\rm d}y {\rm d}t. \end{aligned}$$

(3.28)

A priori estimates are obtained by taking the scalar product in V _3h of each side of (3.27) with $6\Delta t\mathbf{U}^{\tau}_{h}$. We find (comparing with (2.31))

$$\begin{aligned} \begin{aligned} \big|\mathbf{U}^\tau_h\big|^2 - \big|\mathbf{U}^{\tau-1}_h\big|^2 + \big| \mathbf{U}^\tau_h -\mathbf{U}^{\tau-1}_h\big|^2 +2\Delta t\bigl(\mathbf{A}_{3h}\mathbf{U}^\tau_{h}, \mathbf{U}^\tau_{h}\bigr) = 2\Delta t\bigl( \mathbf{F}^\tau_h, \mathbf{U}^\tau_h \bigr), \end{aligned} \end{aligned}$$

and in view of Lemma 3.2 (for A _3h ),

$$\begin{aligned} \big|\mathbf{U}^\tau_h\big|^2 &\leq\big| \mathbf{U}^{\tau-1}_h\big|^2 + 2\Delta t\big| \mathbf{F}^\tau_h\big| \big|\mathbf{U}^\tau_h\big| \\ &\leq\Delta t\big|\mathbf{U}^\tau_h\big|^2 +\big| \mathbf{U}^{\tau-1}_h\big|^2 + \Delta t\big| \mathbf{F}^\tau_{h}\big|^2, \\ \big|\mathbf{U}^\tau_h\big|^2 &\leq\frac{1}{1-\Delta t} \bigl[\big|\mathbf{U}^{\tau -1}_h\big|^2 + \big| \mathbf{F}^\tau_h\big|^2\bigr] \\ &\leq\frac{1}{1-\Delta t} \biggl[ \big|\mathbf{U}^{\tau- 1}_h\big|^2 + \int^{\tau\Delta t}_{(\tau-1)\Delta t} \big|\mathbf{f} ( \cdot , t)\big|^2_{L^2}{\rm d}t \biggr]. \end{aligned}$$

Thus, for $\Delta t\leq\frac{1}{2}$,

$$\begin{aligned} \big|\mathbf{U}^\tau_h\big|^2 \leq4^{\Delta t} \biggl[ \big|\mathbf{U}_h^{\tau -1}\big|^2 + \int^{\tau\Delta t}_{(\tau-1)\Delta t} \big|\mathbf{f}( \cdot ,t)\big|^2_{L^2}{\rm d}t \biggr]. \end{aligned}$$

(3.29)

We write Eq. (3.29) for τ=n+s+1, s=1,…,q. We multiply the equation for τ=n+s+1 by 4^(q−s)Δt and add these equations for s=1,…,q. We find

$$\begin{aligned} \big|\mathbf{U}^{n+q+1}_h\big|^2 \leq4^{q\Delta t} \biggl[ \big|\mathbf{U}^{n+1}_h\big|^2 + \int^{(n+q+1)\Delta t}_{(n+1)\Delta t} \big|\mathbf{f}(\cdot,t)\big|^2_{L^2}{ \rm d}t \biggr]. \end{aligned}$$

(3.30)

During the steps from (n+1)Δt to (n+q+1)Δt, the Z _h are frozen. Thus

$$\begin{aligned} \mathbf{Z}^{n+s +1}_{h} = \mathbf{Z}^{n+1}_h, \quad s =1, \ldots, q, \end{aligned}$$

(3.31)

and we recover the $\mathbf{u}^{n+s +1}_{h}$ in the form

$$\begin{aligned} \mathbf{u}^{n+s+1}_h = \mathbf{U}^{n+s+1}_h +\mathbf{Z}^{n+1}_h. \end{aligned}$$

(3.32)

Then, because of (3.30) and (2.15),

$$\begin{aligned} \big|\mathbf{u}^{n+q+1}_h\big|^2 \leq4^{q\Delta t} \biggl[ \big|\mathbf{u}^{n+1}_h\big|^2 + \int^{(n+q+1)\Delta t}_{(n+1)\Delta t} \big|\mathbf{f}(\cdot,t)\big|^2_{L^2}{ \rm d}t \biggr] . \end{aligned}$$

(3.33)

Combining (3.33) with (3.26), we find

$$\begin{aligned} \begin{aligned} \big|\mathbf{u}^{n+q+1}_h\big|^2 \leq4^{(q+1)\Delta t} \biggl[ \big|\mathbf{u}^n_h\big|^2 + \int^{(n+q+1)\Delta t}_{n\Delta t} \big|\mathbf{f}(\cdot,t)\big|^2_{L^2}{ \rm d}t \biggr] . \end{aligned} \end{aligned}$$

(3.34)

We can repeat the procedure for any interval of time (nΔt,(n+q+1)Δt), n=1,…,N _q , and arrive at the stability result

$$\begin{aligned} \big|\mathbf{u}^m_h\big|^2 & \leq4^{m\Delta t} \biggl[ \big|\mathbf{u}^0_h\big|^2 +\int^{m\Delta t}_0\big|\mathbf{f}(\cdot, t)\big|^2_{L^2}{\rm d}t \biggr] \\ & \leq4^{T} \biggl[ \big|\mathbf{u}^0\big|^2 +\int ^T_0\big|\mathbf{f} (\cdot ,t)\big|^2_{L^2}{ \rm d}t \biggr] \end{aligned}$$

(3.35)

valid for m=1,…,N _q .

Theorem 3.1

The multilevel scheme defined by Eqs. (3.22) and (3.27) is stable in $L^{\infty}(0,T;L^{2}(\mathcal{M})^{3})$ in the sense of (3.35).

4 Other Schemes and Other Methods

The coarse grid schemes that we have used in Sects. 2 and 3 amount to using the same schemes on the coarse grid as on the fine grid. Another possibility for the coarse grid is to average on each coarse grid the fine grid equations associated with the corresponding fine grids. These schemes are made explicit below. However, the study of the stability of these new schemes appears difficult, and we will only present the study of stability in the one-dimensional case for the simple transport equation (see Sect. 4.1), and for a one-dimensional shallow water equation (see Sect. 4.2). Furthermore, the boundary condition will be space periodicity, and the stability analysis is made by the von Neumann method (see [22]).

4.1 The One-Dimensional Case

We start with the one-dimensional space, and consider the same problem as (2.1), with f=0,

$$\begin{aligned} \frac{\partial u}{\partial t}(x,t)+\frac{\partial u}{\partial x}(x,t)=0 \end{aligned}$$

(4.1)

for (x,t)∈(0,L)×(0,T), and with the space periodicity boundary condition, and the initial condition

$$\begin{aligned} u(x,0)=u^0(x). \end{aligned}$$

(4.2)

On the fine grid, we will perform an approximation by the implicit Euler scheme in time and upwind finite-volume in space, so that the scheme will be very much like the one in (2.16) except that the second formula of (2.16) is replaced by the periodicity condition

$$\begin{aligned} u_0^{n+\frac{s+1}{p}} = u_{3N}^{n+\frac{s+1}{p}}. \end{aligned}$$

(4.3)

We perform p steps with a time step $\frac{\Delta t}{p}$ and a space mesh $\Delta x=\frac{L}{3N}$. Then as explained below, we make q steps with a time step Δt and a mesh step 3Δx. Thus we start again with the p steps.

4.1.1 The Fine Grid Scheme with a Small Time Step

The scheme reads

$$\begin{aligned} \frac{p}{\Delta t}\bigl(u^\tau_j-u^{\tau-\frac{1}{p}}_j \bigr)+\frac{1}{\Delta x}\bigl(u^\tau _j-u^\tau_{j-1} \bigr)=0, \end{aligned}$$

(4.4)

where $\tau=n+\frac{s}{p}$, s=1,…,p, j=1,…,3N, $u^{\tau}_{j}$ is meant to be an approximation of $\frac{1}{\Delta x}\int_{k_{j}}u(x,\tau\Delta t){\rm d}x$ with k _j =((j−1)h,jh) and h=Δx; $u^{\tau}_{0}=u^{\tau}_{3N}$ by periodicity.

We associate with a sequence v _j , and its Fourier transform (see [22, p. 38]) is as follows:

$$\begin{aligned} \hat{v}(\xi)=\frac{1}{2\pi}\sum_{j=-\infty}^{+\infty}{ \rm e}^{-{\rm i} j h \xi}v_j h. \end{aligned}$$

(4.5)

Below we will consider periodic sequences v _j , $j\in\mathbb{Z}$, v _j+3N =v _j , $h^{*}=\frac{2\pi}{3N}$ and define the discrete Fourier coefficients (see [5, 9, 22])

$$\begin{aligned} \hat{v}_m=\frac{1}{3N}\sum _{j=1}^{3N}{\rm e}^{- {\rm i}mjh^*}v_j, \quad m=1,\ldots,3N. \end{aligned}$$

(4.6)

We then have the discrete Parseval formula

$$\begin{aligned} \sum_{m=1}^{3N}|\hat{v}_m|^2=\frac{1}{3N}\sum _{j=1}^{3N}|v_j|^2 \end{aligned}$$

(4.7)

(see the details in [3, 22]). Note that the sequence $\{ \hat{v}_{m}\}$ is itself periodic with period 3N, and if (σv) _j =v _j−1, then

$$\begin{aligned} \hat{\sigma v}_m={\rm e}^{-{\rm i}mh^*}\hat{v}_m. \end{aligned}$$

(4.8)

Then (4.4) is rewritten as

$$\begin{aligned} \biggl( 1+\frac{\Delta t}{p\Delta x} \biggr)u^\tau_j- \frac{\Delta t}{p\Delta x}u^\tau _{j-1}=u_j^{\tau-\frac{1}{p}}, \end{aligned}$$

(4.9)

that is, for the Fourier transforms defined as in (4.6), where $h^{*}=\frac{2\pi}{3N}$,

$$\begin{aligned} \biggl( 1+\frac{\Delta t}{p\Delta x}\bigl(1-{\rm e}^{-{\rm i}mh^*}\bigr) \biggr)\hat {u}^\tau _m=\hat{u}^{\tau-\frac{1}{p}}_m, \quad m=1,\ldots,3N. \end{aligned}$$

(4.10)

Hence the amplification factor for the fine mesh is

$$\begin{aligned} g_{F,m}= \biggl[ 1+\frac{\Delta t}{p\Delta x}\bigl(1-{\rm e}^{-{\rm i}mh^*}\bigr) \biggr]^{-1}, \quad m=1,\ldots,3N. \end{aligned}$$

(4.11)

We observe that

$$\begin{aligned} g_{F,m}^{-1}& = \biggl[ 1+\frac{\Delta t}{p\Delta x}\bigl(1-\cos \bigl(h^*m\bigr)\bigr)+{\rm i}\frac{\Delta t}{p\Delta x}\sin \bigl(h^*m\bigr) \biggr], \\ |g_{F,m}^{-1}|^2 & = \biggl[ 1+ \frac{\Delta t}{p\Delta x}\bigl(1-\cos \bigl(h^*m\bigr)\bigr) \biggr]^2+ \biggl(\frac{\Delta t}{p\Delta x} \biggr)^2\sin^2\bigl(h^*m\bigr), \\ & = 1+2\bigl(1-\cos\bigl(h^*m\bigr)\bigr) \biggl( \biggl(\frac{\Delta t}{p\Delta x} \biggr)^2+\frac {\Delta t }{p\Delta x} \biggr). \end{aligned}$$

We conclude that

$$\begin{aligned} |g_{F,m}|\leq1, \quad m=1,\ldots,3N. \end{aligned}$$

(4.12)

Recall that $\tau=n+\frac{s}{p}$, s=1,…,p. Denoting by $u^{\tau}_{h}$ the piecewise constant function given by $u_{h}^{\tau}=u^{\tau}_{j}$ on k _j , (4.7) and (4.12) yield

$$\begin{aligned} \begin{aligned} \big|u_h^{n+\frac{s}{p}}\big|^2&=\sum _{j=1}^{3N}\Delta x\big|u_j^{n+\frac {s}{p}}\big|^2=3N \Delta x\sum_{m=1}^{3N}\big| \hat{u}_m^{n+\frac{s}{p}}\big|^2 \\ & \leq3N\Delta x\sum_{m=1}^{3N}\big| \hat{u}_m^{n}\big|^2=\big|u_h^{n}\big|^2 \quad {\rm for}\ s=1,\ldots,q. \end{aligned} \end{aligned}$$

In particular, for s=p,

$$\begin{aligned} \big|u_h^{n+1}\big|^2 \leq\big|u_h^{n}\big|^2, \end{aligned}$$

(4.13)

and therefore these steps of the scheme (4.4) on the fine grid are stable for the L ²-norm.

4.1.2 The Coarse Grid Scheme with a “Large” Time Step

Considering first the analogue of (4.4) with a time step Δt and a space mesh Δx, we would write (τ=n+s+1 now, s=1,…,q)

$$\begin{aligned} \frac{1}{\Delta t}\bigl(u^\tau_j-u^{\tau-1}_j \bigr)+\frac{1}{\Delta x}\bigl(u^\tau _j-u^\tau_{j-1} \bigr)=0. \end{aligned}$$

(4.14)

To obtain the scheme with a time step Δt and a space mesh 3Δx, we add (average) Eq. (4.14) corresponding to j=3l,3l−1,3l−2.

Setting

$$\begin{aligned} U^\tau_l=\frac{1}{3} \bigl(u^\tau_{3l}+u^\tau_{3l-1}+u^\tau_{3l-2} \bigr), \end{aligned}$$

(4.15)

we obtain

$$\begin{aligned} \frac{1}{\Delta t}\bigl(U^\tau_l-U^{\tau-1}_l \bigr)+\frac{1}{3\Delta x}\bigl(u^\tau _{3l}-u^\tau_{3l-3} \bigr)=0 \end{aligned}$$

(4.16)

for l=1,…,N.

We elaborate on the u=U+Z decomposition (independent of the time step).

The u=U+Z Decomposition

Given the sequence u _j , j=1,…,3N (u ₀=u _3N ), we define the sequence

$$\begin{aligned} U_\ell=\frac{1}{3}\sum _{\alpha=0}^{2}u_{3l-\alpha},\quad l=1,\ldots ,N, \end{aligned}$$

(4.17)

and the sequences

$$\begin{aligned} Z_{3l-\alpha}=u_{3l-\alpha}-U_l, \end{aligned}$$

(4.18)

α=0,1,2, ℓ=1,…,N. We observe that

$$\sum_{\alpha=0}^2Z_{3l-\alpha}=0. $$

Now the multistep algorithm that we consider consists in freezing the Z during the step n+2,…,n+q+1, that is,

$$\begin{aligned} Z^{n+s+1}_j=Z^{n+1}_j, \quad s=1,\ldots,q, \quad j=1,\ldots,3N, \end{aligned}$$

(4.19)

so that

$$Z^\tau_{3l-\alpha}=Z^{n+1}_{3l-\alpha}=u^\tau_{3l-\alpha}-U^\tau_l $$

for α=0,1,2, τ=n+s+1, s=1,…,q. Hence $U^{\tau}_{l}-U^{\tau-1}_{l}=u^{\tau}_{3\ell-\alpha} - u^{\tau-1}_{3\ell-\alpha}$ for α=0,1,2, and for those values of τ. With α=0, (4.16) becomes

$$\begin{aligned} \frac{1}{\Delta t}\bigl(u^\tau_{3l}-u^{\tau-1}_{3l} \bigr)+\frac{1}{3\Delta x}\bigl(u^\tau _{3l}-u^\tau_{3l-3} \bigr)=0. \end{aligned}$$

(4.20)

That is, as in (4.9),

$$\begin{aligned} \biggl(1+\frac{\Delta t}{3\Delta x} \biggr)u^\tau_{3l}- \frac{\Delta t}{3\Delta x}u^\tau _{3l-3}=u^{\tau-1}_{3l}. \end{aligned}$$

(4.21)

Before we introduce the Fourier transform of (4.21) and the amplification function similar to the g _F , we have to elaborate a bit more on the u=U+Z decomposition at the level of the Fourier transforms.

We write (independent of the time step τ), with $h^{*}=\frac{2\pi}{3N}$ for m=1,…,3N,

$$\begin{aligned} \hat{u}_m & =\frac{1}{3N}\sum^{3N}_{j=1} u_j{\rm e}^{-{\rm i}h^*jm} \\ & = \frac{1}{3N}\sum^{N}_{\ell=1} \bigl(u_{3l}{\rm e}^{-3{\rm i} h^*lm}+u_{3l-1}{\rm e}^{-{\rm i}h^*(3l-1)m}+u_{3l-2}{\rm e}^{-{\rm i} h^*(3l-2)m} \bigr). \end{aligned}$$

We now introduce the partial Fourier sum of the type of (4.6),

$$\begin{aligned} \hat{u}_{(3l-\alpha),m}= \frac{1}{3N}\sum _{\ell=1}^{N} u_{3l-\alpha }{\rm e}^{-{\rm i}h^*3lm}. \end{aligned}$$

(4.22)

We observe that this partial Fourier sum is periodic in m with period 3N, and that Parseval relation similar to (4.7) holds,

$$\begin{aligned} \sum_{m=1}^{3N}| \hat{u}_{(3l-\alpha),m}|^2=\frac{1}{3N}\sum _{\ell =1}^{N}|u_{3l-\alpha}|^2,\quad \alpha=0,1,2. \end{aligned}$$

(4.23)

We can hence write

$$\begin{aligned} \hat{u}_m=\hat{u}_{(3l),m}+{\rm e}^{{\rm i}h^*m} \hat{u}_{(3l-1),m}+{\rm e} ^{2{\rm i}h^*m}\hat{u}_{(3l-2),m}. \end{aligned}$$

(4.24)

Then

$$\begin{aligned} \hat{u}_{(3l-3),m}=\hat{u}_{(3l),m}{\rm e}^{-3{\rm i}h^*m}, \end{aligned}$$

(4.25)

and now (4.21) yields by a partial Fourier transform

$$\begin{aligned} \biggl(1+\frac{\Delta t}{3\Delta x} \biggr)\hat{u}^\tau_{(3l),m}- \frac {\Delta t }{3\Delta x}{\rm e}^{-3{\rm i}h^*m}\hat{u}^\tau_{(3l),m}= \hat{u}^{\tau -1}_{(3l),m}. \end{aligned}$$

(4.26)

That is,

$$\begin{aligned} \hat{u}^\tau_{(3l),m}=g_{C,m} \hat{u}^{\tau-1}_{(3l),m}, \quad m=1,\ldots,3N, \end{aligned}$$

(4.27)

corresponding to the amplification factor g _C,m with

$$\begin{aligned} g_{C,m}^{-1}=1+\frac{\Delta t}{3\Delta x}\bigl(1-{\rm e}^{-3{\rm i}h^*m}\bigr). \end{aligned}$$

(4.28)

We can conclude as before that $|g_{C,m}^{-1}|\geq1$,

$$\begin{aligned} |g_{C,m}|\leq1, \quad m=1,\ldots,3N, \end{aligned}$$

(4.29)

and thus the scheme (4.21), (4.26) is “stable”. Also

$$\begin{aligned} \hat{u}^{n+s+1}_{(3l),m}=g_{C,m}^{s} \hat{u}_{(3l),m}^{n+1}, \quad m=1,\ldots,3N, \ s=1,\ldots,q. \end{aligned}$$

(4.30)

The important point now is that we know nothing about the stability of the $u^{\tau}_{3l-1}$, $u^{\tau}_{3l-2}$, and we have to elaborate more to prove this stability.

In the similar way to (4.24), we write for m=1,…,3N,

$$\begin{aligned} \hat{Z}_m=\hat{Z}_{(3l),m}+{\rm e}^{{\rm i}h^*m} \hat{Z}_{(3l-1),m}+{\rm e} ^{2{\rm i}h^*m}\hat{Z}_{(3l-2),m}. \end{aligned}$$

(4.31)

The relations

$$u_{3l-\alpha}=U_l+Z_{3l-\alpha},\quad\alpha=0,1,2 $$

given by the partial discrete Fourier transform for m=1,…,3N,

$$\begin{aligned} \hat{u}_{(3l-\alpha),m}=\hat{U}_{(l),m}+\hat{Z}_{(3l-\alpha),m}. \end{aligned}$$

(4.32)

Hence with (4.19) and (4.32),

$$\begin{aligned} \hat{u}^{n+s+1}_{(3l-\alpha),m}=\hat{U}^{n+s+1}_{(l),m}+ \hat {Z}^{n+1}_{(3l-\alpha),m}. \end{aligned}$$

(4.33)

Using (4.30), we obtain the expression of $\hat{U}^{n+s+1}_{(l),m}$ for α=0,

$$\begin{aligned} \hat{U}_{(l),m}^{n+s+1}=g_{C,m}^{s} \hat{u}^{n+1}_{(3l),m}-\hat {Z}_{(3l),m}^{n+1}, \quad m=1,\ldots,3N. \end{aligned}$$

(4.34)

There remains to express $\hat{Z}_{(3l)}^{n+1}$ in terms of the $\hat{u}^{n+1}_{(3l-\alpha),m}$, α=0,1,2.

We proceed in the physical space, independent of the time step τ, to have

$$U_l=\frac{1}{3}(u_{3l}+u_{3l-1}+u_{3l-2}) $$

and

$$\begin{aligned} \begin{cases} Z_{3l }=u_{3l }-U_l=\frac{1}{3}(2u_{3l }-u_{3l-1}-u_{3l-2}),\\ Z_{3l-1}=u_{3l-1}-U_l=\frac{1}{3}(2u_{3l-1}-u_{3l }-u_{3l-2}),\\ Z_{3l-2}=u_{3l-2}-U_l=\frac{1}{3}(2u_{3l-2}-u_{3l }-u_{3l-1}). \end{cases} \end{aligned}$$

(4.35)

Thus for the Fourier transforms, for m=1,…,3N,

$$\begin{aligned} \begin{cases} \hat{Z}_{(3l),m}=\frac{1}{3}(2\hat{u}_{(3l),m}-\hat{u}_{(3l-1),m}-\hat {u}_{(3l-2),m}),\\ \hat{Z}_{(3l-1),m}=\frac{1}{3}(2\hat{u}_{(3l-1),m}-\hat {u}_{(3l),m}-\hat{u}_{(3l-2),m}),\\ \hat{Z}_{(3l-2),m}=\frac{1}{3}(2\hat{u}_{(3l-2),m}-\hat {u}_{(3l),m}-\hat{u}_{(3l-1),m}). \end{cases} \end{aligned}$$

(4.36)

This holds in particular at the time step τ=n+1.

Now we look for the expression of the $\hat{u}^{n+s+1}_{(3l-\alpha),m}$ (α=0,1,2), in terms of the $\hat{u}^{n+1}_{(3l-\beta),m}$, that of $\hat{u}_{(3l),m}^{n+s+1}$ has been already found (see (4.30)).

By (4.32)–(4.34), (4.36) and (4.19),

$$\begin{aligned} \hat{u}^{n+s+1}_{(3l-1),m}&=\bigl(g_{C,m}^{s}-1 \bigr)\hat{u}^{n+1}_{(3l),m}+\hat {u}^{n+1}_{(3l-1),m}, \end{aligned}$$

(4.37)

$$\begin{aligned} \hat{u}^{n+s+1}_{(3l-2),m}&=\bigl(g_{C,m}^{s}-1 \bigr)\hat{u}^{n+1}_{(3l),m}+\hat {u}^{n+1}_{(3l-2),m}. \end{aligned}$$

(4.38)

We rewrite (4.30), (4.37)–(4.38) in matrical form,

$$\begin{aligned} \begin{pmatrix} \hat{u}^{n+s+1}_{(3l),m}\\ \hat{u}^{n+s+1}_{(3l-1),m}\\ \hat{u}^{n+s+1}_{(3l-2),m} \end{pmatrix} & =G_{C,m}^{(s)} \begin{pmatrix} \hat{u}^{n+1}_{(3l),m}\\ \hat{u}^{n+1}_{(3l-1),m}\\ \hat{u}^{n+1}_{(3l-2),m} \end{pmatrix} ,\quad m=1,\ldots,3N, \end{aligned}$$

(4.39)

$$\begin{aligned} G_{C,m}^{(s)}&= \left(\begin{array}{c@{\quad}c@{\quad}c} g_{C,m}^{s} & 0 & 0\\ g_{C,m}^{s}-1 & 1 &0\\ g_{C,m}^{s}-1 & 0 & 1 \end{array}\right) . \end{aligned}$$

(4.40)

The passing from u ⁿ⁺¹ to u ^n+s+1 is given in the matrical form by (4.39). The stability of the scheme for passing from u ⁿ⁺¹ to u ^n+s+1 is equivalent to showing that the spectral radius of $G^{(s)}_{C,m}$ is not larger than 1 for m=1,…,3N. The eigenvalues of $G_{C,m}^{(s)}$ are not larger than 1. These eigenvalues are 1, 1, $g^{s}_{C,m}$, and we have seen that |g _C,m |≤1.

More precisely, using that the spectral radius of $G_{C,m}^{(s)}$ is less than 1 and (4.23), we have

$$\begin{aligned} \big|u_h^{n+s+1}\big|^2&=\sum _{\alpha=0}^2\sum_{\ell=1}^{N} \Delta x\big|u_{3\ell -\alpha}^{n+s+1}\big|^2=3N\Delta x\sum _{\alpha=0}^2\sum_{m=1}^{3N}\big| \hat {u}_{(3\ell-\alpha),m}^{n+s+1}\big|^2 \\ & \leq 3N\Delta x\sum_{\alpha=0}^2\sum _{m=1}^{3N}\big|\hat{u}_{(3\ell-\alpha ),m}^{n+1}\big|^2=\big|u_h^{n+1}\big|^2 \end{aligned}$$

(4.41)

and for s=q,

$$\begin{aligned} \big|u_h^{n+q+1}\big| \leq\big|u_h^{n+1}\big|. \end{aligned}$$

(4.42)

Combining (4.13) and (4.42), we obtain the stability of the scheme.

Theorem 4.1

The multilevel scheme defined by Eqs. (4.4) and (4.16) is stable in $L^{\infty}(0,\infty;L^{2}(\mathcal{M}))$. More precisely, for all n,

$$\begin{aligned} \big|u_h^n\big|\leq\big|u^0\big|. \end{aligned}$$

(4.43)

4.2 The Linearized 1D Shallow Water Equation

By restriction to 1 dimension, Eq. (3.2) with f=0 become

$$\begin{aligned} \begin{cases} \frac{\partial u}{\partial t}+\tilde{u}_0\frac{\partial u}{\partial x}+g\frac {\partial\phi}{\partial x}=0,\\ \frac{\partial\phi}{\partial t}+\tilde{u}_0\frac{\partial\phi}{\partial x}+\tilde{\phi}_0\frac{\partial u}{\partial x}=0. \end{cases} \end{aligned}$$

(4.44)

We assume the background flow $(\tilde{u}_{0},\tilde{\phi}_{0})$ to be supersonic (supercritical), that is,

$$\begin{aligned} \tilde{u}_0>\sqrt{g\tilde{\phi}_0}. \end{aligned}$$

(4.45)

The boundary conditions are space periodicity, and the initial conditions are given such that they are similar as (3.4). The time and space meshes are the same as in Sects. 2.1 and 2.2.

4.2.1 The Fine Grid Scheme with a “Small” Time Step

The fine grid mesh scheme reads

$$\begin{aligned} \begin{cases} \frac{p}{\Delta t}(u^\tau_j-u^{\tau-\frac{1}{p}}_j)+\frac{\tilde{u}_0}{\Delta x}(u^\tau_j-u^\tau _{j-1})+\frac{g}{\Delta x}(\phi ^\tau_j-\phi^\tau_{j-1})=0,\\ \frac{p}{\Delta t}(\phi^\tau_j-\phi^{\tau-\frac{1}{p}}_j)+\frac{\tilde{u}_0}{\Delta x}(\phi^\tau_j-\phi^\tau _{j-1})+\frac{\tilde{\phi}_0}{\Delta x }(u^\tau_j-u^\tau_{j-1})=0, \end{cases} \end{aligned}$$

(4.46)

where $\tau=n+\frac{s}{p}$, s=1,…,p, j=1,…,3N, $u^{\tau}_{0}=u^{\tau}_{3N}$, $\phi^{\tau}_{0}=\phi^{\tau}_{3N}$ by space periodicity.

We rewrite (4.46) in the form

$$\begin{aligned} \begin{cases} (1+\frac{\tilde{u}_0}{p}\frac{\Delta t}{\Delta x} )u^\tau_j-\frac{\tilde{u}_0 }{p}\frac{\Delta t}{\Delta x}u^{\tau}_{j-1}+\frac{g}{p}\frac{\Delta t}{\Delta x}(\phi ^\tau_j-\phi^\tau_{j-1})=u^{\tau-\frac{1}{p}}_j,\\ (1+\frac{\tilde{u}_0}{p}\frac{\Delta t}{\Delta x} )\phi ^\tau_j-\frac{\tilde{u}_0 }{p}\frac{\Delta t}{\Delta x}\phi^\tau_{j-1}+\frac{\tilde{\phi }_0}{p}\frac{\Delta t}{\Delta x }(u^\tau_j-u^\tau_{j-1})=\phi^{\tau-\frac{1}{p}}_j. \end{cases} \end{aligned}$$

(4.47)

From this, we deduce for the Fourier transforms, for m=1,…,3N,

$$\begin{aligned} \begin{cases} (1+\frac{\tilde{u}_0}{p}\frac{\Delta t}{\Delta x}(1-{\rm e}^{-{\rm i}mh^*}) )\hat{u}_m^\tau+\frac{g}{p}\frac{\Delta t}{\Delta x}\hat{\phi}^\tau _m(1-{\rm e} ^{-{\rm i}mh^*})=\hat{u}^{\tau-\frac{1}{p}}_m,\\ (1+\frac{\tilde{u}_0}{p}\frac{\Delta t}{\Delta x}(1-{\rm e}^{-{\rm i} mh^*}) )\hat{\phi}_m^\tau+\frac{\tilde{\phi}_0}{p}\frac {\Delta t}{\Delta x}\hat {u}_m^\tau(1-{\rm e}^{-{\rm i} mh^*})=\hat{\phi}_m^{\tau-\frac{1}{p}}, \end{cases} \end{aligned}$$

(4.48)

that is,

$$\begin{aligned} \begin{pmatrix} \hat{u}^\tau_m \\ \hat{\phi}_m^\tau \end{pmatrix} = G_{F,m} \begin{pmatrix} \hat{u}^{\tau-\frac{1}{p}}_m \\ \hat{\phi}_m^{\tau-\frac{1}{p}} \end{pmatrix} \end{aligned}$$

(4.49)

with

$$G_{F,m}^{-1}= \left(\begin{array}{c@{\quad}c} 1+\frac{\tilde{u}_0}{p}\frac{\Delta t}{\Delta x}(1-{\rm e}^{-{\rm i}mh^*}) & \frac{g}{p}\frac{\Delta t}{\Delta x}(1-{\rm e}^{-{\rm i} mh^*})\\ \frac{\tilde{\phi}_0}{p}\frac{\Delta t}{\Delta x}(1-{\rm e}^{-{\rm i}mh^*}) & 1+\frac{\tilde{u}_0}{p}\frac{\Delta t}{\Delta x}(1-{\rm e}^{-{\rm i}mh^*}) \end{array}\right) . $$

The eigenvalues of $G_{F,m}^{-1}$ are easily computed

$$\rho_{\pm,m}=1+\varLambda_{\pm}\bigl(1-{\rm e}^{-{\rm i}mh^*} \bigr) $$

with

$$\varLambda_{\pm}=\frac{1}{p} (\tilde{u}_0\pm \sqrt {g\tilde{\phi}_0} )\frac {\Delta t}{\Delta x}. $$

We have

$$|\rho_{\pm,m}|^2=1+2\bigl(1-\cos\bigl(h^*m\bigr)\bigr) \bigl(\varLambda_{\pm}^2+\varLambda _{\pm}\bigr). $$

The condition $\tilde{u}_{0}>\sqrt{g\tilde{\phi}_{0}}$ implies Λ _±>0, and thus

$$|\rho_{\pm,m}|\geq1, \quad m=1,\ldots,3N. $$

Hence, setting u=(u,ϕ) (comparing with (4.13)), we have

$$\begin{aligned} \big|\mathbf{u}^{n+1}_h\big|^2 \leq\big| \mathbf{u}_h^n\big|^2, \end{aligned}$$

(4.50)

so that these steps of the small step scheme (4.46) are stable.

4.2.2 The Coarse Grid Scheme with a “Large” Time Step

We define the cell averages

$$U_l=\frac{1}{3}(u_{3l}+u_{3l-1}+u_{3l-2}), $$

$$\varPhi_l=\frac{1}{3}(\phi_{3l}+ \phi_{3l-1}+\phi_{3l-2}) $$

and the incremental unknowns

$$Z^u_{3l-\alpha}=u_{3l-\alpha}-U_l, $$

$$Z^\phi_{3l-\alpha}=\phi_{3l-\alpha}-\varPhi_l. $$

The analogue of scheme (4.16) reads

$$\begin{aligned} \begin{cases} \frac{1}{\Delta t}(U^\tau_l-U^{\tau-1}_l)+\frac{\tilde {u}_0}{3\Delta x}(u^\tau _{3l}-u^\tau_{3l-3})+\frac{g}{3\Delta x}(\phi^\tau_{3l}-\phi^\tau _{3l-3})=0,\\ \frac{1}{\Delta t}(\varPhi^\tau_l-\varPhi^{\tau-1}_l)+\frac {\tilde{u}_0 }{3\Delta x}(\phi^\tau_{3l}-\phi^\tau_{3l-3})+\frac{\tilde {\phi}_0}{3\Delta x}(u^\tau _{3l}-u^\tau_{3l-3})=0 \end{cases} \end{aligned}$$

(4.51)

for τ=n+s+1, s=1,…,q and l=1,…,N.

Observing as in (4.19) that

$$\begin{aligned} Z_j^{u,n+s+1}=Z^{u,n+1}_j, \quad\quad Z_j^{\phi,n+s+1}=Z^{\phi,n+1}_j \end{aligned}$$

(4.52)

for s=1,…,q, j=1,…,3N and thus that

$$U^\tau_l-U^{\tau-1}_l=u^\tau_{3l}-u^{\tau-1}_{3l}, $$

$$\varPhi^\tau_l-\varPhi^{\tau-1}_l= \phi^\tau_{3l}-\phi^{\tau-1}_{3l}, $$

(4.51) yields

$$\begin{aligned} \begin{cases} \frac{1}{\Delta t}(u^\tau_{3l}-u^{\tau-1}_{3l})+\frac{\tilde {u}_0}{3\Delta x }(u^\tau_{3l}-u^\tau_{3l-3})+\frac{g}{3\Delta x}(\phi^\tau _{3l}-\phi^\tau _{3l-3})=0,\\ \frac{1}{\Delta t}(\phi^\tau_{3l}-\phi^{\tau-1}_{3l})+\frac {\tilde{u}_0}{3\Delta x }(\phi^\tau_{3l}-\phi^\tau_{3l-3})+\frac{\tilde{\phi }_0}{3\Delta x}(u^\tau _{3l}-u^\tau_{3l-3})=0. \end{cases} \end{aligned}$$

(4.53)

Hence, for the partial Fourier transforms for m=1,…,3N (comparing with (4.27)),

$$\begin{aligned} \begin{pmatrix} \hat{u}^\tau_{(3l),m} \\ \hat{\phi}^\tau_{(3l),m} \end{pmatrix} = G_{C,m} \begin{pmatrix} \hat{u}^{\tau-1}_{(3l),m} \\ \hat{\phi}^{\tau-1}_{(3l),m} \end{pmatrix} \end{aligned}$$

(4.54)

with

$$G_{C,m}^{-1}= \left(\begin{array}{c@{\quad}c} 1+\frac{\tilde{u}_0}{3}\frac{\Delta t}{\Delta x}(1-{\rm e}^{-3{\rm i}h^*m}) & \frac{g}{3}\frac{\Delta t}{\Delta x}(1-{\rm e}^{-3{\rm i} h^*m})\\ \frac{\tilde{\phi}_0}{3}\frac{\Delta t}{\Delta x}(1-{\rm e}^{-3{\rm i}h^*m}) & 1+\frac{\tilde{u}_0}{3}\frac{\Delta t}{\Delta x}(1-{\rm e}^{-3{\rm i}h^*m}) \end{array}\right) , $$

where $G_{C,m}^{-1}$ is very similar to $G_{F,m}^{-1}$, and we prove in the same way that its eigenvalues are larger than or equal to 1 in magnitude.

For the moment, we infer from (4.54) that

$$\begin{aligned} \begin{pmatrix} \hat{u}^{n+s+1}_{(3l),m} \\ \hat{\phi}^{n+s+1}_{(3l),m} \end{pmatrix} = G_{C,m}^{s} \begin{pmatrix} \hat{u}^{n+1}_{(3l),m} \\ \hat{\phi}^{n+1}_{(3l),m} \end{pmatrix} . \end{aligned}$$

(4.55)

Then by (4.55),

$$\begin{aligned} \begin{pmatrix} \hat{U}^{n+s+1}_{(l),m} \\ \hat{\varPhi}^{n+s+1}_{(l),m} \end{pmatrix} &= \begin{pmatrix} \hat{u}^{n+s+1}_{(3l),m} \\ \hat{\phi}^{n+s+1}_{(3l),m} \end{pmatrix} - \begin{pmatrix} \hat{Z}^{u,n+s+1}_{(3l),m} \\ \hat{Z}^{\phi,n+s+1}_{(3l),m} \end{pmatrix} \\ &=G_{C,m}^{s} \begin{pmatrix} \hat{u}^{n+1}_{(3l),m} \\ \hat{\phi}^{n+1}_{(3l),m} \end{pmatrix} - \begin{pmatrix} \hat{Z}^{u,n+1}_{(3l),m} \\ \hat{Z}^{\phi,n+1}_{(3l),m} \end{pmatrix} . \end{aligned}$$

(4.56)

We then need to express $\hat{u}^{n+s+1}_{(3l-\alpha),m}$, $\hat{\phi}^{n+s+1}_{(3l-\alpha),m}$ in terms of $\hat{u}^{n+1}_{(3l-\beta),m}$, $\hat{\phi}^{n+1}_{(3l-\beta),m}$, α=1,2, β=0,1,2. We write as in Eqs. (4.37)–(4.38),

$$\begin{aligned} & \begin{pmatrix} \hat{u}^{n+s+1}_{(3l-1),m} \\ \hat{\phi}^{n+s+1}_{(3l-1),m} \end{pmatrix} = \bigl(G_{C,m}^{s}-I \bigr) \begin{pmatrix} \hat{u}^{n+1}_{(3l),m} \\ \hat{\phi}^{n+1}_{(3l)} \end{pmatrix} + \begin{pmatrix} \hat{u}^{n+1}_{(3l-1),m} \\ \hat{\phi}^{n+1}_{(3l-1),m} \end{pmatrix} , \end{aligned}$$

(4.57)

$$\begin{aligned} & \begin{pmatrix} \hat{u}^{n+s+1}_{(3l-2),m} \\ \hat{\phi}^{n+s+1}_{(3l-2),m} \end{pmatrix} = \bigl(G_{C,m}^{s}-I \bigr) \begin{pmatrix} \hat{u}^{n+1}_{(3l),m} \\ \hat{\phi}^{n+1}_{(3l),m} \end{pmatrix} + \begin{pmatrix} \hat{u}^{n+1}_{(3l-2),m} \\ \hat{\phi}^{n+1}_{(3l-2),m} \end{pmatrix} . \end{aligned}$$

(4.58)

In the end,

$$\begin{aligned} \begin{pmatrix} \hat{u}^{n+s+1}_{(3l),m} \\ \hat{\phi}^{n+s+1}_{(3l),m} \\ \hat{u}^{n+s+1}_{(3l-1),m} \\ \hat{\phi}^{n+s+1}_{(3l-1),m} \\ \hat{u}^{n+s+1}_{(3l-2),m} \\ \hat{\phi}^{n+s+1}_{(3l-2),m} \end{pmatrix} = \mathcal{G}_{C,m}^{(s)} \begin{pmatrix} \hat{u}^{n+1}_{(3l),m} \\ \hat{\phi}^{n+1}_{(3l),m} \\ \hat{u}^{n+1}_{(3l-1),m} \\ \hat{\phi}^{n+1}_{(3l-1),m} \\ \hat{u}^{n+1}_{(3l-2),m} \\ \hat{\phi}^{n+1}_{(3l-2),m} \end{pmatrix} , \quad m=1,\ldots,3N \end{aligned}$$

(4.59)

with

$$\mathcal{G}_{C,m}^{(s)} = \left(\begin{array}{c@{\quad}c@{\quad}c} G_{C,m}^{s} & 0 & 0 \\ G_{C,m}^{s}-I & I & 0 \\ G_{C,m}^{s}-I & 0 & I \end{array}\right) . $$

All the eigenvalues of $\mathcal{G}_{C,m}^{(s)}$ are less than or equal to 1, which ensures the stability of the scheme (4.51) going from t=(n+1)Δt to t=(n+s+1)Δt.

Then we have

$$\begin{aligned} \big|\mathbf{u}^{n+s+1}_h\big|\leq\big| \mathbf{u}^{n+1}_h\big|,\quad{\rm for}~ s=1,\ldots,q. \end{aligned}$$

(4.60)

Theorem 4.2

The multilevel scheme defined by Eqs. (4.46) and (4.51) is stable in $L^{\infty}(0,\infty;L^{2}(\mathcal{M})^{2})$. More precisely, for all n,

$$\begin{aligned} \big|\mathbf{u}^{n}_h\big|\leq\big|\mathbf{u}^{0}\big|. \end{aligned}$$

(4.61)