Error Analysis and the Role of Permutation in Dynamic Iteration Schemes

Abstract

As an extension of previous work, we demonstrate that implicit integration methods combined with dynamic iteration schemes applied to selectively determined permutations of differential equations give rise to approximate solutions that converge faster than those obtained using the variable order method. The appropriate permutations strictly depend on the values of the parameters of the given differential system. However, due to the complex influence of the individual parameters on the convergence of the iterations, the relationship between the permutations and the convergence of the resulting schemes remained unknown for systems of higher dimension. In this chapter, we relax this restriction and expand the analysis of the iteration errors and their direct dependence on the given model parameters. Our theoretical analysis of the errors of the dynamic iteration schemes is complemented by illustrative numerical experiments.