WENO schemes for multidimensional nonlinear degenerate parabolic PDEs

Abstract

In this paper, a scheme is presented for approximating solutions of non linear degenerate parabolic equations which may contain discontinuous solutions. In the one-dimensional case, following the idea of the local discontinu ous Galerkin method, first the degenerate parabolic equation is considered as a nonlinear system of first order equations, and then this system is solved us ing a fifth-order finite difference weighted essentially nonoscillatory (WENO) method for conservation laws. This is the first time that the minmod-limiter combined with weighted essentially nonoscillatory procedure has been applied to the degenerate arabolic equations. Also, it is necessary to mention that the new scheme has fifth-order accuracy in smooth regions and second-order accuracy near singularities. The accuracy, robustness, and high-resolution properties of the new scheme are demonstrated in a variety of multidimen sional problems.

Keywords

• weno schemes
• finite difference scheme
• multidimensional nonlinear degenerate parabolic equation
• porous medium equation