The MOL solution of time dependent partial differential equations

A modified BDF scheme is proposed for the numerical integration of the ordinary differential equations that arise in the method of lines solution of time dependent partial differential equations. It is to be expected that this new approach will be superior to the use of BDF for at least three important classes of problems, namely for advection diffusion problems where advection dominates and here we expect very large gains in efficiency, for problems where high accuracy is required and for problems where function evaluations are very expensive. Some analytic and numerical results are given to confirm these expectations.