WENO schemes for multidimensional nonlinear degenerate parabolic PDEs

In this paper, a scheme is presented for approximating solutions of non- linear degenerate parabolic equations which may contain discontinuous solu- tions. In the one-dimensional case, following the idea of the local discontinu- ous Galerkin method, rst the degenerate parabolic equation is considered as a nonlinear system of rst order equations, and then this system is solved us- ing a fth-order nite difference weighted essentially nonoscillatory (WENO) method for conservation laws. This is the rst time that the minmod-limiter combined with weighted essentially nonoscillatory procedure has been applied to the degenerate parabolic equations. Also, it is necessary to mention that the new scheme has fth-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.