Entropy consistent, TVD methods with high accuracy for conservation laws

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.