Abstract
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, chp+ 1y(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 the step size control selects the next step, based on error information from previously accepted steps and the recent step size history. In variable step size implementations the error model must therefore be dynamic and include past step ratios, even in the asymptotic regime. In this paper we derive dynamic asymptotic models of the local error and its estimator, and show how to use dynamically compensated step size controllers that keep the asymptotic local error near a prescribed tolerance tol. The new error models enable the use of controllers with enhanced stability, producing more regular step size sequences. Numerical examples illustrate the impact of dynamically compensated control, and that the proper choice of error estimator affects efficiency.
Highlights
We shall consider a standard initial value problem y = f (t, y), y(0) = y0, Extended author information available on the last page of the article.Numerical Algorithms (2021) 86:537–563 and let yn denote the numerical approximation to y(tn)
We have yn+1 − zn+1 = [Pn+1(tn+1) − y(tn+1)] − [Pn(tn+1) − y(tn+1)] ≈ (cy − cz(ρn)) · hpny+1 · y(py+1)(tn), obtaining a raw error estimate, en = yn+1 − zn+1 ≈ |cy (1) − cz(1)| · y(py+1)(tn) · hpny+1 · πe(ρn), where πe(ρ) accounts for the dynamics of the estimator, with which the controller interacts. This function is derived in the same way as before, by obtaining a variable step size formula for the method, as well as a formula for how the previous polynomial predicts zn+1 through extrapolation
Four well-known standard problems, linear as well as nonlinear, were chosen to benchmark controller performance when different two-step methods were combined with two different types of error estimators
Summary
Extended author information available on the last page of the article. Numerical Algorithms (2021) 86:537–563 and let yn denote the numerical approximation to y(tn). Combinations of these techniques have been applied to near-Hamiltonian problems with weak Rayleigh damping, [16] This theory has focused on one-step methods, for which the asymptotic error model (2) applies whenever the tolerance is small enough. While the theory makes it possible to find method-specific dynamic error estimators in terms of current and past step sizes, the implementation has so far only offered controllers for the static model (2). The main result is that we construct a dynamic compensator that extracts the static part of the asymptotic error model from the estimator, allowing us to employ standard digital filters and other controllers to achieve results comparable to those of onestep methods. For example, the elementary controller (3) is unsuitable for the control of adaptive linear multistep methods, as the stability margin of the process deteriorates with increasing step size history dependence
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