Adaptive WENO Schemes for Singular in Space and Time Solutions of Nonlinear Degenerate Reaction-Diffusion Problems

The numerical solution of nonlinear degenerate reaction-diffusion problems often meets two kinds of difficulties: singularities in space - finite speed of propagation of compact supports' initial perturbations and possible sharp moving fronts, where the solution has low regularity, and singularities in time - blow-up or quenching in finite time. We propose and implement a combination of the sixth-order WENO scheme of Liu, Shu and Zhang [SIAM J. Sci. Comput. 33, 939-965 ( 2011)] with an adaptive procedure to deal with these singularities. Numerical results on the mathematical model of heat structures are shown.