Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Convergence of a generalized Riemann problem scheme for the Burgers equation

Maria Lukacova-Medvidova, Yuhuan Yuan

TL;DR

This work addresses the convergence of a second-order finite-volume scheme for scalar hyperbolic conservation laws, focusing on Burgers equation on a periodic domain with flux $f(u)=\frac{u^2}{2}$ and entropy pair $\eta(u)=\frac{u^2}{2}$, $q(u)=\frac{u^3}{3}$. The authors first show that the GRP-based flux can be entropy-unstable in the presence of shocks and then introduce a stabilized GRP scheme by adding artificial viscosity to enforce a discrete entropy inequality. They prove consistency and convergence of the stabilized scheme to the unique weak entropy solution under a uniform boundedness assumption, using a measure-valued (Young measure) framework and a weak-strong uniqueness argument to obtain strong convergence on the lifespan of any strong solution. The results extend existing Lax-Wendroff-type analyses by establishing convergence via discrete entropy dissipation instead of relying on total-variation growth, thus providing a robust, entropy-stable multi-dimensional GRP-based method for Burgers-type equations and potentially broader scalar conservation laws.

Abstract

In this paper we study the convergence of a second order finite volume approximation of the scalar conservation law. This scheme is based on the generalized Riemann problem (GRP) solver. We firstly investigate the stability of the GRP scheme and find that it might be entropy unstable when the shock wave is generated. By adding an artificial viscosity we propose a new stabilized GRP scheme. Under the assumption that numerical solutions are uniformly bounded, we prove consistency and convergence of this new GRP method.

Convergence of a generalized Riemann problem scheme for the Burgers equation

TL;DR

This work addresses the convergence of a second-order finite-volume scheme for scalar hyperbolic conservation laws, focusing on Burgers equation on a periodic domain with flux and entropy pair , . The authors first show that the GRP-based flux can be entropy-unstable in the presence of shocks and then introduce a stabilized GRP scheme by adding artificial viscosity to enforce a discrete entropy inequality. They prove consistency and convergence of the stabilized scheme to the unique weak entropy solution under a uniform boundedness assumption, using a measure-valued (Young measure) framework and a weak-strong uniqueness argument to obtain strong convergence on the lifespan of any strong solution. The results extend existing Lax-Wendroff-type analyses by establishing convergence via discrete entropy dissipation instead of relying on total-variation growth, thus providing a robust, entropy-stable multi-dimensional GRP-based method for Burgers-type equations and potentially broader scalar conservation laws.

Abstract

In this paper we study the convergence of a second order finite volume approximation of the scalar conservation law. This scheme is based on the generalized Riemann problem (GRP) solver. We firstly investigate the stability of the GRP scheme and find that it might be entropy unstable when the shock wave is generated. By adding an artificial viscosity we propose a new stabilized GRP scheme. Under the assumption that numerical solutions are uniformly bounded, we prove consistency and convergence of this new GRP method.
Paper Structure (30 sections, 12 theorems, 105 equations, 1 figure, 1 table)

This paper contains 30 sections, 12 theorems, 105 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Numerical method
  3. Notations
  4. Mesh.
  5. Function space.
  6. Discrete differential operators.
  7. Reconstruction operator.
  8. Time discretization.
  9. Minmod limiter
  10. Finite volume GRP method
  11. Stability
  12. Properties of the "RP" flux
  13. Case 1 -- $\left\llbracket R_V u_h \right\rrbracket < 0,~ \left\{\space\left\{ R_V u_h \right\} \space\right\} > 0$:
  14. Case 2 -- $\left\llbracket R_V u_h \right\rrbracket < 0,~ \left\{\space\left\{ R_V u_h \right\} \space\right\} \leq 0$:
  15. Case 3 -- $0< ( R_V u_h )^{-}_{\sigma} < ( R_V u_h )^{+}_{\sigma}$:
  16. ...and 15 more sections

Key Result

Lemma 3.1

For a numerical solution $u_h$ obtained by a finite volume method with an RP flux $f_{\sigma}^{RP}(R_V u_h)$ we have

Figures (1)

  • Figure 1: Structured mesh with $d = 2$.

Theorems & Definitions (25)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Remark 3.3
  • Lemma 3.4: Entropy balance
  • proof
  • Lemma 3.5: Uniform bounds
  • ...and 15 more