Implicit Methods on Parallel Processors

Since most partial differential equations (PDEs) do not have exact solutions, they are usually solved by some type of numerical method. Since a numerical method is commonly built from finite difference approximations derived from Taylor series expansions, such a development is derived. Stability and convergence of these methods is defined and the rate of convergence is defined and shown for a few simple methods. Of particular importance is the difference between implicit and explicit methods. Finally, the current applications and adaptations of implicit methods on parallel processors are examined and their strengths and weaknesses discussed