Numerical solution of parabolic differential equations using finite differences: a comparative study

Abstract

In this work, the finite difference method was applied to parabolic differential equations in order to compare numerical results of problems involving thermal diffusion. The numerical solution is obtained by using the explicit, implicit and implicit numerical methods of Crank-Nicolson. The stability and consistency criteria of each method are described. The numerical results showed that the implicit formulations outweigh the stability conditions of the explicit method, making it possible to use larger steps in the mesh. The approximations, however, are more satisfactory when those steps tend to zero. Taking into account the convergence criteria, the formulations presented in this paper present reliable numerical solutions of the problems. The implicit and Crank-Nicolson methods are considerably better in relation to the explicit method.