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On The Weak Consistency of Finite Volumes Schemes for Conservation Laws on General Meshes

Thierry Gallouët, R. Herbin, J. -C Latché

TL;DR

This work develops a weak consistency framework (Lax-Wendroff type) for conservative finite-volume schemes solving $\\partial_t u + \\nabla \\cdot F(u)=0$ on general, unstructured meshes. It introduces a discrete gradient operator $∇_{\\mathcal E}$ and proves its convergence to the continuous gradient in $L^ fty$-weak$^*$ under a uniform mesh-regularity bound, along with a result showing that discrete translations of $L^1$ functions vanish in the limit. Using these tools, the authors establish a weak consistency theorem: if the discrete solutions converge in $L^1$ to $\\bar u$ and the fluxes converge appropriately with a Lip-diagonal bound, then $\\bar u$ satisfies the weak formulation of the conservation law, including the initial condition. The framework extends Lax-Wendroff-type arguments to general meshes with minimal regularity assumptions, guiding the design and analysis of finite-volume schemes for multi-dimensional hyperbolic systems and ensuring convergence to weak solutions.

Abstract

The aim of this paper is to develop some tools in order to obtain the weak consistency of (in other words, analogues of the Lax-Wendroff theorem for) finite volume schemes for balance laws in the multi-dimensional case and under minimal regularity assumptions for the mesh. As in the seminal Lax-Wendroff paper, our approach relies on a discrete integration by parts of the weak formulation of the scheme. This makes a discrete gradient of the test function appear, and the central argument for the scheme consistency is to remark that this discrete gradient is convergent in L $\infty$ weak .

On The Weak Consistency of Finite Volumes Schemes for Conservation Laws on General Meshes

TL;DR

This work develops a weak consistency framework (Lax-Wendroff type) for conservative finite-volume schemes solving on general, unstructured meshes. It introduces a discrete gradient operator and proves its convergence to the continuous gradient in -weak under a uniform mesh-regularity bound, along with a result showing that discrete translations of functions vanish in the limit. Using these tools, the authors establish a weak consistency theorem: if the discrete solutions converge in to and the fluxes converge appropriately with a Lip-diagonal bound, then satisfies the weak formulation of the conservation law, including the initial condition. The framework extends Lax-Wendroff-type arguments to general meshes with minimal regularity assumptions, guiding the design and analysis of finite-volume schemes for multi-dimensional hyperbolic systems and ensuring convergence to weak solutions.

Abstract

The aim of this paper is to develop some tools in order to obtain the weak consistency of (in other words, analogues of the Lax-Wendroff theorem for) finite volume schemes for balance laws in the multi-dimensional case and under minimal regularity assumptions for the mesh. As in the seminal Lax-Wendroff paper, our approach relies on a discrete integration by parts of the weak formulation of the scheme. This makes a discrete gradient of the test function appear, and the central argument for the scheme consistency is to remark that this discrete gradient is convergent in L weak .

Paper Structure

This paper contains 5 sections, 6 theorems, 58 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Space discretization
  3. A weakly convergent discrete gradient
  4. Convergence of "discrete translations" of functions of $L^1$
  5. Weak consistency of conservative finite volumes discretizations of conservation laws

Key Result

Lemma 3.1

\newlabellem:grad Let $({\mathcal{M}}^{(m)})_{m \in \mathbb{N}}$ be a sequence of meshes such that the mesh step $h_{{\mathcal{M}}^{(m)}}$ tends to zero when $m$ tends to $+\infty$. We suppose that the mesh parameters $\theta^\nabla_{{\mathcal{M}}^{(m)}}$ defined by eq:deftheta1 are uniformly bound Let $\varphi \in C_c^\infty(\Omega)$ and, for $m \in \mathbb{N}$, let ${\boldsymbol \nabla}_{{\mathc

Figures (2)

  • Figure 2.1: Mesh and associated notations.
  • Figure 3.1: Choice of ${\boldsymbol x}_K$ and ${\boldsymbol x}_L$ for "flat cells": (a) convenient choice, (b) quasi-orthogonality is lost when cells become flatter and flatter.

Theorems & Definitions (17)

  • Lemma 3.1
  • proof
  • Remark 3.1
  • Remark 3.2: Choice of ${\boldsymbol x}_K$ and $D_\sigma$
  • Remark 3.3: On the mesh regularity assumption
  • Lemma 3.2
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 7 more