Positivity preserving and unconditionally stable numerical scheme for the three-dimensional modified Fisher–Kolmogorov–Petrovsky–Piskunov equation

Abstract

This paper introduces a numerical approach for the practical solution of the modified Fisher–Kolmogorov–Petrovsky–Piskunov equation that describes population dynamics. The diffusion term and nonlinear term is based on the operator splitting method and interpolation method, respectively. The analytic proof of the discrete maximum principle and positivity preserving for the numerical algorithm is demonstrated. Numerical solution calculated using the proposed method remains stable without blowing up, which implies that the proposed method is unconditionally stable. Numerical studies show that the proposed method is second-order convergence in space and first-order convergence in time. The performance and applicability of the proposed scheme are studied through various computational tests that present the effects of model parameters and evolution dynamics.