Adaptive linear equation solvers in codes for large stiff systems of odes

Abstract

Iterative linear equation solvers have been shown to be effective in codes for large systems of stiff initial-value problems for ordinary differential equations (ODEs). While preconditioned iterative methods are required in general for efficiency and robustness, unpreconditioned methods may be cheaper over some ranges of the interval of integration. In this paper, we develop a strategy for switching between unpreconditioned and preconditioned iterative methods depending on the amount of work being done in the iterative solver and properties of the matrix being solved. This strategy is combined with a "type-insensitive" approach to the choice of formula used in the ODE code to develop a method that makes a smooth transition between nonstiff and stiff when the type-insensitive approach is used. If there is a region of the integration that is "mildly" stiff, switchign between unpreconditioned and preconditioned iterative methods also increases the efficiency of the code significantly.