A second-order accurate finite-difference scheme for the classical Fisher-Kolmogorov-Petrovsky-Piscounov equation

Abstract

In this paper, we introduce a wide family of practical numerical methods with two weighing parameters to solve the classical Fisher-Kolmogorov-Petrovsky-Piscounov equation from population dynamics. Our approach is based on the classical Crank-Nicholson method combined with method of lagging to obtain temporally and spatially second-order accurate numerical estimates. Indeed, for a special value of parameters both equal to half we prove that the scheme is second-order accurate in time and in space. In this case, stability, consistency, and (therefore) convergence of the method are examined and it is shown that the method is unconditionally stable under a non-restrictive condition. Two numerical examples are presented and compared with the exact analytical solutions for their order of convergence