Numerical Solution to Parabolic PDE Using Implicit Finite Difference Approach

Abstract

This paper examines an implicit Finite Difference approach for solving the parabolic partial differential equation (PDE) in one dimension. We consider the Crank Nicolson scheme which offers a better truncation error for both time and spatial dimensions as compared with the explicit Finite Difference method. In addition the scheme is consistent and unconditionally stable. One downside of implicit methods is the relatively high computational cost involved in the solution process, however this is compensated by the high level of accuracy of the approximate solution and efficiency of the numerical scheme. A physical problem modelled by the heat equation with Neumann boundary condition is solved using the Crank Nicolson scheme. Comparing the numerical solution with the analytical solution, we observe that the relative error increases sharply at the right boundary, however it diminishes as the spatial step size approaches zero.

Keywords: Partial Differential Equation, Implicit Finite Difference, Crank Nicolson Scheme