Compact Finite Difference Scheme Combined with Richardson Extrapolation for Fisher’s Equation

Hailu Muleta Chemeda,Ayana Deressa Negassa,Feyisa Edosa Merga,Francisco Urena

doi:10.1155/2022/7887076

Abstract

In this study, the fourth-order compact finite difference scheme combined with Richardson extrapolation for solving the 1D Fisher’s equation is presented. First, the derivative involving the space variable is discretized by the fourth-order compact finite difference method. Then, the nonlinear term is linearized by the lagging method, and the derivative involving the temporal variable is discretized by the Crank–Nicolson scheme. The method is found to be unconditionally stable and fourth-order accurate in the direction of the space variable and second-order accurate in the direction of the temporal variable. When combined with the Richardson extrapolation, the order of the method is improved from fourth to sixth-order accurate in the direction of the space variable. The numerical results displayed in figures and tables show that the proposed method is efficient, accurate, and a good candidate for solving the 1D Fisher’s equation.

Highlights

Mathematical modeling of most physical systems leads to linear/nonlinear partial differential equations (PDEs) in various fields of science
PDEs have enormous applications compared to ordinary differential equations (ODEs) such as in dynamics, electricity, heat transfer, electromagnetic theory, quantum mechanics, and so on [1]
To validate the performance of the proposed method, we considered two numerical examples whose exact solution is available. e pointwise absolute error ε at t T is approximated by ε 􏼌􏼌􏼌􏼌uexact xi, T􏼁 − uapprox xi, T􏼁􏼌􏼌􏼌􏼌, 1 ≤ i ≤ N − 1

Summary

Introduction

Mathematical modeling of most physical systems leads to linear/nonlinear partial differential equations (PDEs) in various fields of science. Bastani and Salkuyeh [14] had combined a CFD6 scheme for second derivative in space and a third-order total variation diminishing the Runge–Kutta (TVD-RK3) scheme in time to approximate Fisher’s equation. Another way for improving the accuracy and rate of convergence of the FDM is through the application of Richardson’s extrapolation (RE) provided that their error term is expressible as a polynomial or power series in h [21, 22]. We establish the stability condition and order of convergence of the proposed method

Mathematical Formulation and Analysis of the Proposed Method

Results, Discussion, and Conclusion

Method x