A moving mesh method for one-dimensional hyperbolic conservation laws

John M. Stockie, John A. MacKenzie, Robert D. Russell

Research output: Contribution to journalArticle

68 Citations (Scopus)

Abstract

We develop an adaptive method for solving one-dimensional systems of hyperbolic conservation laws that employs a high resolution Godunov-type scheme for the physical equations, in conjunction with a moving mesh PDE governing the motion of the spatial grid points. Many other moving mesh methods developed to solve hyperbolic problems use a fully implicit discretization for the coupled solution-mesh equations, and so suffer from a significant degree of numerical stiffness. We employ a semi-implicit approach that couples the moving mesh equation to an efficient, explicit solver for the physical PDE, with the resulting scheme behaving in practice as a two-step predictor-corrector method. In comparison with computations on a fixed, uniform mesh, our method exhibits more accurate resolution of discontinuities for a similar level of computational work.
LanguageEnglish
Pages1791-1813
Number of pages23
JournalSIAM Journal on Scientific Computing
Volume22
Issue number5
DOIs
Publication statusPublished - 2001

Fingerprint

Moving Mesh Method
Hyperbolic Conservation Laws
Moving Mesh
Conservation
Stiffness
Mesh
Predictor-corrector Methods
Two-step Method
Hyperbolic Problems
Semi-implicit
One-dimensional System
Adaptive Method
Discontinuity
High Resolution
Discretization
Grid
Motion

Keywords

  • moving mesh
  • adaptivity
  • equidistribution
  • shock capturing
  • hyperbolic conservation laws
  • finite volume methods
  • numerical mathematics

Cite this

Stockie, John M. ; MacKenzie, John A. ; Russell, Robert D. / A moving mesh method for one-dimensional hyperbolic conservation laws. In: SIAM Journal on Scientific Computing. 2001 ; Vol. 22, No. 5. pp. 1791-1813.
@article{80477a03f16548af9b626577de6fc39e,
title = "A moving mesh method for one-dimensional hyperbolic conservation laws",
abstract = "We develop an adaptive method for solving one-dimensional systems of hyperbolic conservation laws that employs a high resolution Godunov-type scheme for the physical equations, in conjunction with a moving mesh PDE governing the motion of the spatial grid points. Many other moving mesh methods developed to solve hyperbolic problems use a fully implicit discretization for the coupled solution-mesh equations, and so suffer from a significant degree of numerical stiffness. We employ a semi-implicit approach that couples the moving mesh equation to an efficient, explicit solver for the physical PDE, with the resulting scheme behaving in practice as a two-step predictor-corrector method. In comparison with computations on a fixed, uniform mesh, our method exhibits more accurate resolution of discontinuities for a similar level of computational work.",
keywords = "moving mesh, adaptivity, equidistribution, shock capturing, hyperbolic conservation laws, finite volume methods, numerical mathematics",
author = "Stockie, {John M.} and MacKenzie, {John A.} and Russell, {Robert D.}",
year = "2001",
doi = "10.1137/S1064827599364428",
language = "English",
volume = "22",
pages = "1791--1813",
journal = "SIAM Journal on Scientific Computing",
issn = "1064-8275",
number = "5",

}

A moving mesh method for one-dimensional hyperbolic conservation laws. / Stockie, John M.; MacKenzie, John A.; Russell, Robert D.

In: SIAM Journal on Scientific Computing, Vol. 22, No. 5, 2001, p. 1791-1813.

Research output: Contribution to journalArticle

TY - JOUR

T1 - A moving mesh method for one-dimensional hyperbolic conservation laws

AU - Stockie, John M.

AU - MacKenzie, John A.

AU - Russell, Robert D.

PY - 2001

Y1 - 2001

N2 - We develop an adaptive method for solving one-dimensional systems of hyperbolic conservation laws that employs a high resolution Godunov-type scheme for the physical equations, in conjunction with a moving mesh PDE governing the motion of the spatial grid points. Many other moving mesh methods developed to solve hyperbolic problems use a fully implicit discretization for the coupled solution-mesh equations, and so suffer from a significant degree of numerical stiffness. We employ a semi-implicit approach that couples the moving mesh equation to an efficient, explicit solver for the physical PDE, with the resulting scheme behaving in practice as a two-step predictor-corrector method. In comparison with computations on a fixed, uniform mesh, our method exhibits more accurate resolution of discontinuities for a similar level of computational work.

AB - We develop an adaptive method for solving one-dimensional systems of hyperbolic conservation laws that employs a high resolution Godunov-type scheme for the physical equations, in conjunction with a moving mesh PDE governing the motion of the spatial grid points. Many other moving mesh methods developed to solve hyperbolic problems use a fully implicit discretization for the coupled solution-mesh equations, and so suffer from a significant degree of numerical stiffness. We employ a semi-implicit approach that couples the moving mesh equation to an efficient, explicit solver for the physical PDE, with the resulting scheme behaving in practice as a two-step predictor-corrector method. In comparison with computations on a fixed, uniform mesh, our method exhibits more accurate resolution of discontinuities for a similar level of computational work.

KW - moving mesh

KW - adaptivity

KW - equidistribution

KW - shock capturing

KW - hyperbolic conservation laws

KW - finite volume methods

KW - numerical mathematics

U2 - 10.1137/S1064827599364428

DO - 10.1137/S1064827599364428

M3 - Article

VL - 22

SP - 1791

EP - 1813

JO - SIAM Journal on Scientific Computing

T2 - SIAM Journal on Scientific Computing

JF - SIAM Journal on Scientific Computing

SN - 1064-8275

IS - 5

ER -