Carlos D. Acosta, Raimund Bürger:
Difference schemes stabilized by discrete mollification for degenerate parabolic equations in two space dimensions
The discrete mollification method is a convolution-based filtering procedure for the regularization of ill-posed problems. This method is applied here to stabilize explicit schemes, which were first analyzed by Karlsen & Risebro [M2AN Math. Model. Numer. Anal. 35 (2001), 239-269], for the solution of initial value problems of strongly degenerate parabolic PDEs in two space dimensions. Two new schemes are proposed, which are based on direction-wise and two-dimensional discrete mollification of the second partial derivatives forming the Laplacian of the diffusion function, respectively. The mollified schemes permit to use substantially larger time steps than the original (basic) scheme. It is proven that both schemes converge to the unique entropy solution of the initial value problem. Numerical examples demonstrate that the mollified schemes are competitive in efficiency, and in many cases significantly more efficient, than the basic scheme.
This preprint gave rise to the following definitive publication(s):
Carlos D. ACOSTA, Raimund BüRGER: Difference schemes stabilized by discrete mollification for degenerate parabolic equations in two space dimensions. IMA Journal of Numerical Analysis, vol. 32, 4, pp. 1509-1540, (2012).