A paradigm for well-balanced schemes for traveling waves emerging in parabolic biological models

We propose a methodology for designing well-balanced numerical schemes to investigate traveling waves in parabolic models from mathematical biology. We combine well-balanced techniques for parabolic models known in the literature with the so-called LeVeque-Yee formula as a dynamic estimate for the spreading speed. This latter formula is used to consider the evolution problem in a moving frame at each time step, where the equations admit stationary solutions, for which well-balanced techniques are suitable. Then, the solution is shifted back to the stationary frame in a well-balanced manner. We illustrate this methodology on parabolic reaction-diffusion equations, such as the Fisher/Kolmogorov-Petrovsky-Piskunov Equation, and a class of equations with a cubic reaction term that exhibit a transition from pulled to pushed waves. We show that the numerical schemes capture in a consistent way simultaneously the wave speed and, to an extent, the so-called Bramson delay.