A new highly accurate fourth-order approximation for Fisher equation

Abstract

Fisher Equation (FE) is a nonlinear partial differential equation which is used to model many physical systems involving effects of both linear diffusions and nonlinear reactions. FE arises in numerous applications including brain tumor dynamics, population dynamics, chemical reactions, etc. Finite Difference Approximation (FDA) method has widely been used to obtain discrete solutions of FE. Combined with Crank-Nicolson (C-N) technique, some second and fourth order accurate FDA’s were also derived in the literature. In these methods, the nonlinear part of FE is discretized to its corresponding linear form using the lagging technique. In this study, a new fourth-order C-N scheme is developed for FE with a new linearization technique. Our approach is two-fold. First, a pre-conditioned operator (𝑃ℎ) for the second derivative (𝐷2) is constructed. The pre-conditioned 𝐷2 (𝑃ℎ𝐷2) is computed with order 4 accuracy, using the order 2 central FDA for 𝐷2. Second, the nonlinear part of FE is discretized at the middle point of two consecutive time steps with the order 2 accuracy using concepts of arithmetic and geometric means. The resulting C-N scheme (CN1) is second-order accurate in time and fourth-order accurate in space. Furthermore, using Richard extrapolation, the accuracy of the spatial variable is raised to order 6. Numerical results obtained from CN1 and a recently developed order 4 compact C-N scheme (CN2), with and without extrapolation, demonstrated that CN1 is more accurate than CN2, for example, the maximum errors of the extrapolated CN1 and CN2 are 2.764e- 14 and 1.121e-06, respectively for grid steps ℎ = 0.0125(space) and 𝜏 = 0.00015625(time).