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We propose a methodology for designing well-balanced numerical schemes toinvestigate traveling waves in parabolic models from mathematical biology. Wecombine well-balanced techniques for parabolic models known in the literaturewith the so-called LeVeque-Yee formula as a dynamic estimate for the spreadingspeed. This latter formula is used to consider the evolution problem in amoving frame at each time step, where the equations admit stationary solutions,for which well-balanced techniques are suitable. Then, the solution is shiftedback to the stationary frame in a well-balanced manner. We illustrate thismethodology on parabolic reaction-diffusion equations, such as theFisher/Kolmogorov-Petrovsky-Piskunov Equation, and a class of equations with acubic reaction term that exhibit a transition from pulled to pushed waves. Weshow that the numerical schemes capture in a consistent way simultaneously thewave speed and, to an extent, the so-called Bramson delay.

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