SOME STABILITY ASPECTS OF SCHEMES FOR THE ADAPTIVE INTEGRATION OF STIFF INITIAL-VALUE PROBLEMS

The phenomenon of "superstability" in the numerical solution of stiff initial value problems of ordinary differential equations is considered. Superstability occurs when many implicit schemes are used, and here the backward differentiation formulas and the discontinuous Galerkin method are concentrated on. In the latter case, some implementation aspects are discussed. These methods are implemented with "local" and "global" error control strategies, respectively. By comparison, the effect that the different strategies have on the occurrence of superstability in two model problems is studied, and the trapezoidal rule implemented with a local error control strategy is also considered. New insight is gained into the superstability phenomenon, and it becomes clear that the method used to solve the nonlinear algebraic system, usually a (quasi)Newton method, has a large bearing. This problem is also considered in the context of the numerical integration of parabolic partial differential equations by the method of lines. In particular, the consequences for the dynamical behavior of solutions of a reaction-diffusion problem are studied.