### Abstract

A complete self-contained treatment of the stability and convergence properties of conservation-form, monotone difference approximations to scalar conservation laws in several space variables is developed. In particular, the authors prove that general monotone difference schemes always converge and that they converge to the physical weak solution satisfying the entropy condition. Rigorous convergence results follow for dimensional splitting algorithms when each step is approximated by a monotone difference scheme. The results are general enough to include, for instance, Godunovs scheme, the upwind scheme (differenced through stagnation points), and the Lax-Friedrichs scheme together with appropriate multi-dimensional generalizations.

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

Pages (from-to) | 1-21 |

Number of pages | 21 |

Journal | Mathematics of Computation |

Volume | 34 |

Issue number | 149 |

DOIs | |

State | Published - 1980 |

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### ASJC Scopus subject areas

- Algebra and Number Theory
- Computational Mathematics
- Applied Mathematics

### Cite this

*Mathematics of Computation*,

*34*(149), 1-21. https://doi.org/10.1090/S0025-5718-1980-0551288-3

**Monotone difference approximations for scalar conservation laws.** / Crandall, Michael G.; Majda, Andrew.

Research output: Contribution to journal › Article

*Mathematics of Computation*, vol. 34, no. 149, pp. 1-21. https://doi.org/10.1090/S0025-5718-1980-0551288-3

}

TY - JOUR

T1 - Monotone difference approximations for scalar conservation laws

AU - Crandall, Michael G.

AU - Majda, Andrew

PY - 1980

Y1 - 1980

N2 - A complete self-contained treatment of the stability and convergence properties of conservation-form, monotone difference approximations to scalar conservation laws in several space variables is developed. In particular, the authors prove that general monotone difference schemes always converge and that they converge to the physical weak solution satisfying the entropy condition. Rigorous convergence results follow for dimensional splitting algorithms when each step is approximated by a monotone difference scheme. The results are general enough to include, for instance, Godunovs scheme, the upwind scheme (differenced through stagnation points), and the Lax-Friedrichs scheme together with appropriate multi-dimensional generalizations.

AB - A complete self-contained treatment of the stability and convergence properties of conservation-form, monotone difference approximations to scalar conservation laws in several space variables is developed. In particular, the authors prove that general monotone difference schemes always converge and that they converge to the physical weak solution satisfying the entropy condition. Rigorous convergence results follow for dimensional splitting algorithms when each step is approximated by a monotone difference scheme. The results are general enough to include, for instance, Godunovs scheme, the upwind scheme (differenced through stagnation points), and the Lax-Friedrichs scheme together with appropriate multi-dimensional generalizations.

UR - http://www.scopus.com/inward/record.url?scp=84966217981&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84966217981&partnerID=8YFLogxK

U2 - 10.1090/S0025-5718-1980-0551288-3

DO - 10.1090/S0025-5718-1980-0551288-3

M3 - Article

AN - SCOPUS:84966217981

VL - 34

SP - 1

EP - 21

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 149

ER -