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 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 using a fifth—order finite difference weighted essentially nonoscillatory (WEN O) 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 parabolic 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 multidimensional problems.

کلیدواژه ها:

WENO schemes ، Finite difference scheme ، Multidimensional nonlinear degenerate parabolic equation ، Porous medium equation