Convergence of High Order Finite Volume Weighted Essentially Nonoscillatory Scheme and Discontinuous Galerkin Method for Nonconvex Conservation Laws

In this paper, we consider the issue of convergence toward entropy solutions for high order finite volume weighted essentially non-oscillatory (WENO) scheme and discontinuous Galerkin (DG) finite element method approximating scalar nonconvex conservation laws. Although such high order nonlinearly stable schemes can usually converge to entropy solutions of convex conservation laws, convergence may fail for certain nonconvex conservation laws. We perform a detailed study to demonstrate such convergence issues for a few representative examples, and suggest a modification of the high order schemes based either on first order monotone schemes or a second order entropic projection [1] to achieve convergence toward entropy solutions while maintaining high order accuracy in smooth regions.