Nonlinear stability of discrete shocks for systems of conservation laws
type: Item
instance of: scholarly article
title: Nonlinear stability of discrete shocks for systems of conservation laws
zbMATH Open document ID: 0807.35087
DOI: 10.1007/BF00383220
author: Jian-Guo Liu
author: Zhouping Xin
published in: Archive for Rational Mechanics and Analysis
publication date: 1994-02-22T00:00:00Z
review text: The authors study the asymptotic nonlinear stability of discrete shocks for the Lax-Friedrichs difference scheme for systems of nonlinear hyperbolic conservation laws. The main result states that a single weak discrete shock is stable in \(L^ p\) for all \(p \geq 1\) if the initial perturbations sum to zero. Further, it is shown that if the solution of the Riemann problem for the far-field states consists only of shocks, one from each characteristic family, then the corresponding discrete solution is stable in \(L^ p\) for \(p \geq 2\). The methods used include a weighted estimate and the characteristic energy method, based on the essential monotonicity of the Lax-Friedrichs scheme.
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 35L67
zbMATH DE Number: 501116
zbMATH Keywords: finite differences
zbMATH Keywords: nonlinear stability of discrete shocks
zbMATH Keywords: Lax- Friedrichs difference scheme
zbMATH Keywords: Riemann problem
reviewed by: Michael Shearer
MaRDI profile type: Publication
cites work: Large-Time Behavior of Solutions of Lax-Friedrichs Finite Difference Equations for Hyperbolic Systems of Conservation Laws
cites work: Monotone Difference Approximations for Scalar Conservation Laws
cites work: Convergence of approximate solutions to conservation laws
cites work: One-Sided Difference Approximations for Nonlinear Conservation Laws
cites work: Nonlinear asymptotic stability of viscous shock profiles for conservation laws
cites work: Q4011905
cites work: Viscous limits for piecewise smooth solutions to systems of conservation laws
cites work: Rarefactions and large time behavior for parabolic equations and monotone schemes
cites work: Discrete shocks
cites work: Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion
cites work: Weak solutions of nonlinear hyperbolic equations and their numerical computation
cites work: Q3213564
cites work: L 1 -Stability of Stationary Discrete Shocks
cites work: Discrete shocks for difference approximations to systems of conservation laws
cites work: A class of convergent finite difference schemes for certain nonlinear parabolic systems
cites work: Q5536978
cites work: On Convergence of Monotone Finite Difference Schemes with Variable Spatial Differencing
cites work: Existence and stability of stationary profiles of the lw scheme
cites work: On the stability of finite element methods for shock waves
cites work: Nonlinear stability of viscous shock waves
cites work: The Large-Time Behavior of the Scalar, Genuinely Nonlinear Lax-Friedrichs Scheme
cites work: On the linearized stability of viscous shock profiles for systems of conservation laws
full work available at URL: https://doi.org/10.1007/bf00383220
OpenAlex ID: W2043358921
description: scientific article; zbMATH DE number 501116