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Abstract
Finite difference schemes for the Fisher-KPP equation are considered that are constructed using non-local discretization of non-linear terms and standard/nonstandard denominators. Their dynamic consistency with respect to positivity, boundedness, steady-states, and diffusion-front propagation are established. Diffusion-fronts are obtained that inflate, deflate, or remain constant at a geometric rate determined by the time-space step size functional relationship. The persistence of said dynamic consistency in terms of their boundedness, positivity, and determining-diffusion curves at small and large scales is rigorously analyzed and numerically supported. Comparisons are presented for three cases using the Euler, nonstandard finite difference, or exact spectral derivative discretization finite difference denominators. Finally, general measures of robustness in persistence are introduced that incorporate (mesh) scale sizes and are used to rank the robustness of the schemes investigated.
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