Some efficient numerical schemes for approximating the nonlinear two-space dimensional extended Fisher-Kolmogorov equation

Abstract

This paper develops some efficient numerical schemes based on the hybridization of finite difference and orthogonal cubic spline collocation (OCSC) techniques to approximate the nonlinear two-dimensional extended Fisher-Kolmogorov model. This model represents a strong nonlinear fourth-order reaction diffusion evolution model. The proposed strategy is based on second-order backward differentiation formula and θ-scheme (including the Crank-Nicolson and backward Euler methods). It is shown that both the semi-discrete and full discrete approaches are unconditionally stable and convergent by means of the energy analysis procedure in an appropriate Sobolev space. In addition, the optimal convergence rates are obtained including fourth-order and second-order accuracies in space and time directions, respectively. Finally, numerical results assess the accuracy of the proposed methods and verify theoretical estimates.