ABSTRACT

The finite difference method (FDM) seems to be the simplest approach for the numerical solution of PDEs. It proceeds by replacing those derivatives in the governing equations by finite differences. In this chapter, we will introduce various difference methods for parabolic equations. In Sec. 2.1, we present both the explicit and implicit schemes for a simple heat equation. Then we introduce some important concepts (such as stability, consistence, and convergence) used in analyzing finite difference methods in Sec. 2.2. Then in Sec. 2.3, we demonstrate a few examples for using those concepts. In Sec. 2.4, we extend the discussion to two-dimensional and three-dimensional parabolic equations. Here we cover the standard difference methods and the Alternate Direction Implicit (ADI) method. Finally, in Sec. 2.5, we present a MATLAB code to show readers how to solve a parabolic equation.