Computational Stability and Adaptive Strategies - An Experimental Study of ODE Software

Abstract: The algorithmic content of adaptive ODE/DAE software is dominated by a considerable amount of control structures, support algorithms and logic. The control logic has been largely heuristic and lacking support by analysis. In this thesis we study adaptive time-stepping based on control theory, combined with other carefully selected control strategies. Thus we argue that ODE/DAE software can be constructed and analyzed by proven, "standard" scientific techniques instead of heuristics. We show that carefully designed adaptive algorithms have a most significant impact on the reliability and computational stability of ODE/DAE codes. The latter notion implies that minor changes of the computational setup must only lead to small changes in the computed result: the software must be computationally well-conditioned. A series of computational experiments with the standard implementations of DASSL and RADAU5 are compared to results obtained with modified versions of these codes, where the improved strategies build on the theory mentioned above. These include stepsize control based on digital filters. The experiments demonstrate that these "minor" algorithmic changes strongly improve computational stability at no extra computational expense.