Stepsize control in the solution of initial value ODEs is the prime tool to meet with given accuracy requirements as efficiently as possible for a specific time-stepping method. Normally this is accomplished by using asymptotic results of how some local error relates to the stepsize; the stepsize is chosen so that this local error fulfils the prescribed accuracy requirements. In practically all ODE solvers using linear multistep methods (LMMs) the asymptotic expressions are supplemented by several heuristics, favouring a constant stepsize. There are mainly two reasons for this: (1) It is difficult to make a rigorous error and stability analysis for variable stepsize LMMs; to obtain a closer resemblance with the fixed stepsize analysis one tries to restrain stepsize variations and also explicitly use the error estimates obtained from the fixed stepsize analysis for stepsize (and order) control. (2) Several quantities of the LMMs, e.g. method coefficients and data representation, depends on the stepsize history; the overhead due to the computation of these quantities can be reduced, if the stepsize is constant during a number of steps.

Our approach is to treat the stepsize control from a control theoretical point of view, where we work with the error estimates obtained from the variable stepsize analysis, inter alia to be able to bring in the dependency on the stepsize history into the stepsize control. Furthermore, we try to keep the heuristics of the control algorithms to a minimum; by using a better stepsize controller, we should be able to obtain a smoother stepsize sequence, which also brings about a more efficient integration that compensates for the overhead due to e.g. the method coefficients computations.

MINISIMPOSIUM SESSION: 19. Representation and implementation of variable step multistep methods