Differential Equations and Computational Simulations III
Electron. J. Diff. Eqns., Conf. 01, 1997, pp. 181-201.

Entropy consistent, TVD methods with high accuracy for conservation laws

Xuefeng Li

The Godunov method for conservation laws produces numerical solutions that are total-variation diminishing (TVD) and converge to weak solutions which satisfy the entropy condition (Entropy Consistency), but the method is only first order accurate. Many second and higher order accurate Godunov-type methods have been developed by various researchers. Although these high order methods perform very well numerically, convergence and entropy-consistency has not been proven, maybe due to the highly nonlinear approach. In this paper, we develop a new class of Godunov-type methods that are TVD, converge to weak solutions of conservation laws, and satisfy the entropy condition. The error produced by these methods are theoretically controllable by the choice the piecewise constant functions used in the numerical approximation. Numerical experiments confirm that our methods produce numerical solutions that are comparable to those produced by higher order methods, while maintaining all the good characteristics of the Godunov method.

Published November 12, 1998.
Mathematics Subject Classification: 65C20, 65M12, 65M06.
Key words and phrases: Conservation Laws, Godunov Method, Entropy Condition, Convergence, High Accuracy.

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Xuefeng Li
Department of Mathematics and Computer Science
Loyola University
6363 St. Charles Avenue
New Orleans, LA 70118, USA.
E-mail address: Li@Loyno.edu

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