Numerical computation of sharp travelling waves of a degenerate diffusion–reaction equation arising in biofilm modelling

• A numerical method for travelling waves of degenerate diffusion–reaction equations with gradient blow-up at the wave front.
• The underlying idea is to approximate the double degenerate problem by a sequence of single degenerate problems.
• The wave speed of the original problem is found as the limit of the wave speed of these auxiliary problems.
• The method is tested on a model that arises in the study of bacterial biofilms.

Many degenerate diffusion–reaction equations permit sharp travelling wave solutions that describe the propagation of an interface with finite speed. If the equation is at least double degenerate, the derivative of the travelling wave solution can blow up at the interface, which poses considerable challenges for the computation of the travelling wave speed. We propose a numerical method for this problem that is based on the idea to approximate the multiple degenerate problem by a family of simple degenerate problems. For the latter we propose an interval-bracketing algorithm based on the theory of Sanchez-Garduno and Maini. The travelling wave speed of the original problem is obtained as the limit of the travelling wave speeds of the auxiliary problems. The performance of the method is investigated in a numerical simulation experiment for a problem that arises in the mathematical modelling of biofilm processes.