Existence of monostable fronts for a KPP infinite-difference numerical scheme

TL;DR

The paper analyzes traveling fronts for a semi-discretized KPP equation with small spatial step $h$, proving existence of monostable fronts for all speeds $c>2\sqrt{g'(0)}$ when $h$ is sufficiently small. The main novelty is a spectral-convergence framework that transfers invertibility from the continuous front problem to the discrete setting, yielding $h$-uniform resolvent bounds and enabling a fixed-point construction even for infinite-range discretizations with geometrically decaying (possibly signed) kernels. A far-field ansatz and over-localized weighted spaces are employed to control residual terms and to manage the lack of a comparison principle, extending bistable spectral-convergence techniques to monostable problems. The result broadens the class of discretization schemes for which traveling fronts can be established, and paves the way for analysis of slowest waves and fully discrete time schemes in this monostable context.

Abstract

We study the existence of traveling wave solutions for a numerical counterpart of the KPP equation. We obtain the existence of monostable fronts for all super-critical speeds in the regime where the spatial step size is small. The key strategy is to transfer the invertibility of certain linear operators related to the front solutions from the continuous setting to the discrete case we are interested in. We rely on resolvent bounds which are uniform with respect to the step size, a procedure which is also known as spectral convergence. The approach is also able to handle infinite range discretizations with geometrically decaying coefficients that are allowed to have both signs, which prevents the use of the comparison principle.

Theorems & Definitions (34)

• Theorem 2
• Theorem 4
• Lemma 5
• proof
• Lemma 6
• proof
• Lemma 7
• proof
• Lemma 8
• proof