### 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 L^{2} 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|u_{j}
^{n}|≦0.24 is satisfied.

Original language | English (US) |
---|---|

Pages (from-to) | 429-452 |

Number of pages | 24 |

Journal | Numerische Mathematik |

Volume | 30 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1978 |

### Fingerprint

### 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.

Research output: Contribution to journal › Article

*Numerische Mathematik*, vol. 30, no. 4, pp. 429-452. https://doi.org/10.1007/BF01398510

}

TY - JOUR

T1 - A systematic approach for correcting nonlinear instabilities - The Lax-Wendroff scheme for scalar conservation laws

AU - Majda, Andrew

AU - Osher, Stanley

PY - 1978/12

Y1 - 1978/12

N2 - 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.

AB - 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.

KW - Subject Classifications: Primary: 65M10, Secondary: 65M05, 35L65

UR - http://www.scopus.com/inward/record.url?scp=0012323913&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0012323913&partnerID=8YFLogxK

U2 - 10.1007/BF01398510

DO - 10.1007/BF01398510

M3 - Article

AN - SCOPUS:0012323913

VL - 30

SP - 429

EP - 452

JO - Numerische Mathematik

JF - Numerische Mathematik

SN - 0029-599X

IS - 4

ER -