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On the stability of finite-volume schemes on non-uniform meshes

Pavel Bakhvalov, Mikhail Surnachev

TL;DR

This work addresses the problem of ensuring $L_2$-stability for high-order finite-volume schemes solving the 1D transport equation on non-uniform meshes with small periodic perturbations of a uniform grid. It develops a spectral-cell analysis that leverages a block-structure representation and a holomorphic perturbation framework to establish a sufficient stability condition (Theorem th:stab) for meshes in a family $\mathcal{F}_{\mu}$. Building on this, the authors prove $(p+1)$-th order convergence (supra-convergence) for FV schemes with $p=2s$-order polynomial reconstructions, provided the scheme is $(\varkappa-1)$-exact in the sense of a local mapping and the mesh deformation is small enough. The paper also supplements the theory with numerical experiments on checkerboard meshes, showing stable behavior for certain schemes (e.g., second-order polynomial reconstruction) and instability for others (e.g., R3 beyond a threshold and R5 on any non-uniform mesh), thereby validating the stability criterion and the supra-convergence claim with empirical evidence.

Abstract

In this paper, we study the L2 stability of high-order finite-volume schemes for the 1D transport equation on non-uniform meshes. We consider the case when a small periodic perturbation is applied to a uniform mesh. For this case, we establish a sufficient stability condition. This allows to prove the (p+1)-th order convergence of finite-volume schemes based on p-th order polynomials.

On the stability of finite-volume schemes on non-uniform meshes

TL;DR

This work addresses the problem of ensuring -stability for high-order finite-volume schemes solving the 1D transport equation on non-uniform meshes with small periodic perturbations of a uniform grid. It develops a spectral-cell analysis that leverages a block-structure representation and a holomorphic perturbation framework to establish a sufficient stability condition (Theorem th:stab) for meshes in a family . Building on this, the authors prove -th order convergence (supra-convergence) for FV schemes with -order polynomial reconstructions, provided the scheme is -exact in the sense of a local mapping and the mesh deformation is small enough. The paper also supplements the theory with numerical experiments on checkerboard meshes, showing stable behavior for certain schemes (e.g., second-order polynomial reconstruction) and instability for others (e.g., R3 beyond a threshold and R5 on any non-uniform mesh), thereby validating the stability criterion and the supra-convergence claim with empirical evidence.

Abstract

In this paper, we study the L2 stability of high-order finite-volume schemes for the 1D transport equation on non-uniform meshes. We consider the case when a small periodic perturbation is applied to a uniform mesh. For this case, we establish a sufficient stability condition. This allows to prove the (p+1)-th order convergence of finite-volume schemes based on p-th order polynomials.
Paper Structure (10 sections, 12 theorems, 109 equations, 1 figure, 2 tables)

This paper contains 10 sections, 12 theorems, 109 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Problem formulation and the main result
  3. Proof of Theorem \ref{['th:stab']}
  4. The schemes with the polynomial reconstruction
  5. Example: meshes with alternating steps
  6. Scheme based on second-order polynomials
  7. Scheme R3
  8. Scheme R5
  9. Numerical results
  10. Conclusion

Key Result

Theorem 2.1

Consider the scheme eq1, where $a_k$ are holomorphic at $(1, \ldots, 1)$ and satisfy eq_invalpha. Let $\varkappa$ be defined by eq_ring_lambda. Let the scheme Then for each $K>1$ there exists $\mu : \mathbb{N} \to (0,\infty)$ such that for each solution of eq1 on each mesh $X \in \mathcal{F}_{\mu}$, where $\mathcal{F}_{\mu}$ is given by eq_def_fmu, there holds

Figures (1)

  • Figure 1: A checkerboard mesh

Theorems & Definitions (23)

  • Theorem 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • ...and 13 more