An unconditionally stable splitting scheme for a class of nonlinear parabolic equations

We propose and analyse a numerical scheme for a class of advection-dominate d advection-diffusion-reaction equations. The scheme is essentially based o n operator splitting combined with a front tracking method for conservation laws, which tracks an evolving piecewise constant solution with discontinu ity paths defined by a varying velocity field. The splitting separates out the advection, which is modelled by a nonlinear conservation law, and the d iffusion/reaction. Since the front tracking scheme (unlike conventional met hods) has no associated time step, our numerical scheme can be made uncondi tionally stable by choosing appropriate methods for the diffusion and react ion steps. Nevertheless, it is observed that when the time step is notably larger than the diffusion scale, the scheme can become too diffusive. This can be inferred by the fact that the entropy condition forces the hyperboli c solver to throw away information (entropy loss) regarding the structure o f steep fronts. We will demonstrate that the disregarded information can be identified as a residual flux term. Moreover, if this residual flux is tak en into account via, for example, a separate correction step, steep fronts can be given the correct amount of self-sharpening. Four numerical examples are presented. The first three examples discuss the quality of the approxi mate solutions in terms of accuracy and efficiency. The last example is dra wn from glacier modelling.