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

We propose and analyse a numerical scheme for a class of advection-dominated advection-diffusion-reaction equations. The scheme is essentially based on operator splitting combined with a front tracking method for conservation laws, which tracks an evolving piecewise constant solution with discontinuity paths defined by a varying velocity field. The splitting separates out the advection, which is modelled by a nonlinear conservation law, and the diffusion/reaction. Since the front tracking scheme (unlike conventional methods) has no associated time step, our numerical scheme can be made unconditionally stable by choosing appropriate methods for the diffusion and reaction 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 hyperbolic solver to throw away information (entropy loss) regarding the structure of steep fronts. We will demonstrate that the disregarded information can be identified as a residual flux term. Moreover, if this residual flux is taken 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 approximate solutions in terms of accuracy and efficiency. The last example is drawn from glacier modelling.