Adaptive Linear Equation Solvers in Codes for Large Stiff Systems of ODEs

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Abstract
Iterative linear equation solvers have been shown to be efiective in codes for large systemsofstifiinitial-valueproblemsforordinarydifierentialequations(ODEs). Whilepreconditioned iterativemethodsarerequiredingeneralfore-ciencyandrobustness,unpreconditionedmethodsmay be cheaper over some ranges of the interval of integration. In this paper, we develop a strategy for switchingbetweenunpreconditionedandpreconditionediterativemethodsdependingontheamount 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 developamethodthatmakesasmoothtransitionbetweennonstifiandstifiregimesintheintervalof integration. Weflndthat,asexpected,forsomelargesystemsofODEs,theremaybeaconsiderable saving in execution time when the type-insensitive approach is used. If there is a region of the integration that is \mildly" stifi, switching between unpreconditioned and preconditioned iterative methods also increases thee-ciency ofthe code signiflcantly.

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