Electronic Journal of Differential Equations,
Conference 01 (1998), pp. 171-191

Title: Entropy consistent, TVD methods with high accuracy for conservation laws

Authors: Xuefeng Li (Loyola Univ., New Orleans, LA, USA)
 
Abstract: 
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.
Math Subject Classifications: 65C20, 65M12, 65M06
Key Words: Conservation Laws; Godunov Method; Entropy Condition; Convergence; High Accuracy.