Numerical Engineering: Experiences in Designing PDE Software with Selfadaptive Variable Step Size/Variable Order Difference Methods

The basic ideas in designing software for the numerical solution of nonlinear systems of elliptic and parabolic PDE’s with variable step size/variable order difference methods are presented. The error is estimated by the difference of difference formulae, using members of families of difference formulae. Basic solution methods are developed for the solution of the BVP and the IVP for ODE’s. These methods are extended and combined to solution methods for elliptic and parabolic PDE’s. The nonlinear equations are solved by a robust Newton-Raphson method. The method tends to balance all the relevant errors according to a prescribed relative tolerance. For the final solution an estimate of the error of the solution is computed which means e. g. a global error for the IBVP’s.