Explicit and Implicit Difference Methods for Quasilinear First Order Partial Functional Differential Equations

Initial boundary value problems of the Dirichlet type for quasilinear functional differential equations are considered. Explicit difference schemes of the Euler type 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 classes of the methods. It is shown that conditions on the mesh for explicit difference schemes are more restrictive than suitable assumptions for implicit methods. Error estimates for both methods are presented. Interpolating operators corresponding to functional variables are constructed.

High-Order Compact Implicit Difference Methods For Parabolic Equations in Geodynamo Simulation

A series of compact implicit schemes of fourth and sixth orders are developed for solving differential equations involved in geodynamics simulations. Three illustrative examples are described to demonstrate that high-order convergence rates are achieved while good efficiency in terms of fewer grid points is maintained. This study shows that high-order compact implicit difference methods provide high flexibility and good convergence in solving some special differential equations on nonuniform grids.