### 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) |
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Pages (from-to) | 429-452 |

Number of pages | 24 |

Journal | Numerische Mathematik |

Volume | 30 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1 1978 |

### Keywords

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

### ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics