The method of lines for the numerical solution of partial differential equations

The numerical solution of time dependent nonlinear partial differential equations is a complicated and nontrivial process. Generally speaking, the solution of slightly different types of partial differential equations has required the development of separate and different computer programs. This situation for partial differential equations is in direct contrast to that for ordinary differential equations. Recently, sophisticated and reliable computer programs for the automatic solution of nonlinear systems of ordinary differential equations have become available. The use of the method of lines approach for solving partial differential equations is an attempt to take advantage of these recent significant developments for ordinary differential equations. In this presentation we will compare the advantages and disadvantages of this approach. We will describe how the method of lines approach for solving PDE's allows one to be able to develop very general purpose computer software for solving broad classes of problems. The structure and use of this software will be described. Finally, numerical results obtained with this general purpose software will be presented and discussed.