Abstract: We propose and analyse a numerical scheme for a class of advection dominated advection--diffusion--reaction equations. The scheme is essentially based on combining a front tracking method for conservation laws, which tracks shock curves defined by a varying velocity field, with a suitable operator splitting. The splitting is formulated for an equation in non-conservative form and consists of a nonlinear conservation law modelling advection, a heat equation modelling diffusion, and finally an ordinary differential equation modelling lower order processes. Since no CFL condition is associated with the front tracking scheme, our numerical scheme is unconditionally stable in the sense that the splitting time step is not restricted by the spatial discretization parameter. Nevertheless, it is observed that when the splitting time step is notably larger than the diffusion scale, the scheme can become too diffusive. This can be inferred with the fact that the entropy condition forces the hyperbolic solver to throw away information regarding the structure of shock fronts. We will demonstrate that it is possible to identify what is thrown away as a residual flux term. Moreover, if this residual flux is taken into account via, for example, a separate correction step, the shock fronts can be given the correct amount of self sharpening. Two numerical examples are presented and discussed. The first is an academic test case while the second is drawn from glacier modelling.
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