Adaptive Linear Multistep Methods Designing Automatic Step Size Control for Multistep Methods

In a k-step adaptive linear multistep methods the coefficients depend on the k-1 most recent step size ratios. In a similar way, both the actual and the estimated local error will depend on these step ratios. The classical error model has been the asymptotic model, r=c h^{p+1} y^{(p+1)}(t), based on the constant step size analysis, where all past step sizes simultaneously go to zero. This does not reflect actual computations with multistep methods, where step size control only affects future steps, not the previous accepted steps. In variable step size implementations, therefore, even in the asymptotic regime, the error model must include the dependence on previous step sizes and step ratios. In this talk we develop dynamic asymptotic models for variable step size computations, and analyze a few new step size controllers accounting for the dynamics in the error model, while keeping the local error near a prescribed tolerance.