A systematic approach for correcting nonlinear instabilities - The Lax-Wendroff scheme for scalar conservation laws

Andrew Majda, Stanley Osher

Research output: Contribution to journalArticle

Abstract

The Lax-Wendroff (L-W) difference scheme for a single conservation law has been shown to be nonlinearly unstable near stagnation points. In this paper a simple variant of L-W is devised which has all the usual properties-conservation form, three point scheme, second order accurate on smooth solutions, but which is shown rigorously to be L2 stable for Burger's equation and which is believed to be stable in general. This variant is constructed by adding a simple nonlinear viscosity term to the usual L-W operator. The nature of the viscosity is deduced by first stabilizing, for general conservation laws, a model differential equation derived by analyzing the truncation error for L-W, keeping only terms of order (Δt)2. The same procedure is then carried out for an analogous semi-discrete model. Finally, the full L-W difference scheme is rigorously shown to be stable provided that the C.F.L. restriction λ|uj n|≦0.24 is satisfied.

Original languageEnglish (US)
Pages (from-to)429-452
Number of pages24
JournalNumerische Mathematik
Volume30
Issue number4
DOIs
StatePublished - Dec 1978

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Nonlinear Instability
Scalar Conservation Laws
Conservation
Difference Scheme
Conservation Laws
Viscosity
Stagnation Point
Truncation Error
Smooth Solution
Term
Burgers Equation
Discrete Model
Differential equations
Unstable
Differential equation
Restriction
Operator
Model

Keywords

  • Subject Classifications: Primary: 65M10, Secondary: 65M05, 35L65

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics
  • Mathematics(all)

Cite this

A systematic approach for correcting nonlinear instabilities - The Lax-Wendroff scheme for scalar conservation laws. / Majda, Andrew; Osher, Stanley.

In: Numerische Mathematik, Vol. 30, No. 4, 12.1978, p. 429-452.

Research output: Contribution to journalArticle

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