Carlos D. Acosta, Raimund Bürger, Carlos E. Mejia:
Monotone difference schemes stabilized by discrete mollification for strongly degenerate parabolic equations
The discrete mollification method is a convolution-based filtering procedure suitable for the regularization of ill-posed problems and for the stabilization of explicit schemes for the solution of PDEs. This method is applied to the discretization of the diffusive terms of a known first-order monotone finite difference scheme (S. Evje and K.H. Karlsen, SIAM J Numer Anal 37 (2000) 1838--1860) for initial value problems of strongly degenerate parabolic equations in one space dimension. It is proved that the mollified scheme is monotone, and converges to the unique entropy solution of the initial value problem, under a CFL stability condition which permits to use time steps that are larger than with the un-mollified (basic) scheme. Several numerical experiments illustrate the performance, and gains in CPU time, for the mollified scheme. Applications to initial-boundary value problems are included.
This preprint gave rise to the following definitive publication(s):
Carlos D. ACOSTA, Raimund BüRGER, Carlos E. MEJIA: Monotone difference schemes stabilized by discrete mollification for strongly degenerate parabolic equations. Numerical Methods for Partial Differential Equations, vol. 28, 1, pp. 38-62, (2012).