Fourth-Order Compact Finite Difference Method Combined with Richardson Extrapolation for OneDimensional Heat Equation

Abstract

In this paper, Fourth-Order Compact Finite Difference method combined with Richardson extrapolation to solve the one-dimensional heat equation has been presented. The method is found to be fourth-order convergent in space and second-order in time variable. The method is also unconditionally stable and consistent for solving a one-dimensional heat equation. When combined with the Richardson extrapolation, the desired level of numerical solutions can be obtained without the requirement of further mesh refinement which claims extra computational time and memory. To validate the applicability of the proposed method, two model examples are considered and solved for different values of spatial and temporal step lengths. The numerical solutions presented in tables and the simulations of the model problems show that the proposed method approximates the exact solution very well. In a nutshell, the proposed method is efficient and capable of solving the one-dimensional heat equation.