Adaptive mesh point selection for the efficient solution of scalar IVPs

Abstract

We discuss adaptive mesh point selection for the solution of scalar IVPs. We consider a method that is optimal in the sense of the speed of convergence, and aim at minimizing the local errors. Although the speed of convergence cannot be improved by using the adaptive mesh points compared to the equidistant points, we show that the factor in the error expression can be significantly reduced. We obtain formulas specifying the gain achieved in terms of the number of discretization subintervals, as well as in terms of the prescribed level of the local error. Both nonconstructive and constructive versions of the adaptive mesh selection are shown, and a numerical example is given.Comment: 21 pages. Numerical Algorithms final versio

Adaptive mesh selection asymptotically guarantees a prescribed local error for systems of initial value problems

“…We shortly comment on comparison between the algorithm defined in [4] for scalar autonomous problems and the current algorithm designed for systems of IVPs, for the test problem (72). As it can be expected, the algorithm from [4] allows us to better treat small values of δ. This follows from the fact that, roughly speaking, the step size control in [4] was based on two-sided estimates of local errors.…”

“…This follows from the fact that, roughly speaking, the step size control in [4] was based on two-sided estimates of local errors. Specific properties of scalar autonomous problems were used in [4]; they cannot be extended to systems of initial value problems. In order to handle systems of IVPs, the present algorithm uses upper local error bounds, see (28) and (29).…”

“…In the similar spirit, adaption has been considered for univariate approximation and minimization in [1]. For scalar autonomous problems (1), adaptive mesh selection has been recently studied in [4]. An adaptive strategy has been proposed and the cost analyzed, based on specific properties of scalar autonomous equations.…”

“…Theorem 2 Let f ∈ F r , β > 0 and ϕ ∈ (0, 1). There exists m 0 such that for all m ≥ m 0 , for any {x i } m i=0 satisfying (4), and anyx i+1 satisfying (24) it holds…”

Asymptotically tight worst case complexity bounds for initial-value problems with nonadaptive information

“…A closer look however shows that advantages of adaption depend very much on the problem itself and the class of problem instances being solved. It is not a purpose of this paper to discuss the adaption/nonadaption issue in details -to have a flavor of it, one can consult the monograph [9], or recent papers [1], [5], [8].…”