One-sided difference approximations for nonlinear conservation laws

Authors:
Björn Engquist and Stanley Osher

Journal:
Math. Comp. **36** (1981), 321-351

MSC:
Primary 65M10; Secondary 35L67, 65M05

DOI:
https://doi.org/10.1090/S0025-5718-1981-0606500-X

MathSciNet review:
606500

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Abstract: We analyze one-sided or upwind finite difference approximations to hyperbolic partial differential equations and, in particular, nonlinear conservation laws. Second order schemes are designed for which we prove both nonlinear stability and that the entropy condition is satisfied for limit solutions. We show that no such stable approximation of order higher than two is possible. These one-sided schemes have desirable properties for shock calculations. We show that the proper switch used to change the direction in the upwind differencing across a shock is of great importance. New and simple schemes are developed for which we prove qualitative properties such as sharp monotone shock profiles, existence, uniqueness, and stability of discrete shocks. Numerical examples are given.

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DOI:
https://doi.org/10.1090/S0025-5718-1981-0606500-X

Article copyright:
© Copyright 1981
American Mathematical Society