Explicit and Implicit Finite Difference Methods

Following Burden et al. [5], we will consider the parabolic partial differential equation in one space dimension  

As part of the development of a finite difference scheme to approximate the solution to Equation 9.90 to Equation 9.92, we must select two mesh constants, h and k, such that m = Ljh is an integer. The grid points are (r tf), where 

In the important case of no source term, and when both end conditions are zero. Equation 9.94 reduces to 

The method described by Equation 9.94 to Equation 9.98 is an explicit method and is commonly called the forward difference method. It is explicit because knowledge of Wjj for j at all the grid points means that 

The forward difference method is considered conditionally stable. Further, it can be shown that the method converges with a rate of convergence on the order of k + /t ) if the condition 

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In a similar manner the stability of other explicit and implicit finite difference methods may be examined. This has been done by Lapidus and Finder [4], who conclude that most... 

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