ABSTRACT
This chapter provides an overview of the numerical solution of ordinary and partial differential equations (PDE), without attempting to describe all the possible methods and the refinements that are built into the general-purpose optimized software packages. In general, the numerical solution of PDEs requires the solution of very large systems of non-homogeneous linear equations. The numerical schemes used in professional software packages are usually more complicated than the Crank-Nicholson scheme, although they use the same basic approach and are also implicit. Consequently, in practice numerical results are obtained and plotted with the aid of a computer, using one of the efficient and highly optimized commercial software packages that are available. The modification that is often used, particularly when seeking numerical solutions of PDEs, is called the successive over-relaxation process. The numerical solution of PDEs is an important and complicated subject, and detailed descriptions of numerical procedures are to be found in many textbooks.
