Comparison between some explicit and implicit difference schemes for Hamilton Jacobi functional differential equations

Initial boundary value problems of Dirichlet type for first order partial functional differential equations are considered. Explicit difference schemes and implicit difference methods are investigated. Sufficient conditions for the convergence of approximate solutions are given and comparisons of the methods are presented. It is proved that assumptions on the regularity of given functions are the same for both the methods. It is shown that conditions on the mesh for explicit difference schemes are more restrictive than suitable assumptions for implicit methods. There are implicit difference schemes which are convergent and corresponding explicit difference methods are not convergent. Error estimates for both the methods are constructed. Numerical examples are presented.