Comparison of Explicit and Implicit Finite Difference Schemes on Diffusion Equation

Abstract

In physics and mathematics, heat equation is a special case of diffusion equation and is a partial differential equation (PDE). Partial differential equations are useful tools for mathematical modeling. A few problems can be solved analytically, whereas difficult boundary value problem can be solved by numerical methods easily. A very popular numerical method known as finite difference methods (explicit and implicit schemes) is applied expansively for solving heat equations successfully. Explicit schemes are Forward Time and Centre Space (FTCS), Dufort and Frankel methods, whereas implicit schemes are Laasonen and Crank-Nicolson methods. In this study, explicit and implicit finite difference schemes are applied for simple one-dimensional transient heat conduction equation with Dirichlet’s initial-boundary conditions. MATLAB code is used to solve the problem for each scheme in fine mesh grids. Comparing results with analytical results, Crank-Nicolson method gives the best approximate solution. FTCS scheme is conditionally stable, whereas other schemes are unconditionally stable. Convergence, stability and truncation error analysis are investigated. Transient temperature distribution plot and surface temperature plots for different time are presented. Also, unstable plot for FTCS method is represented.