Analysis and Optimization of an Adaptive Interpolation Algorithm for the Numerical Solution of a System of Ordinary Differential Equations with Interval Parameters
A. Yu. Morozov, Andrey Andreevich Zhuravlev, Dmitry Leonidovich Reviznikov · Differential Equations · 2020
Abstract We consider an adaptive interpolation algorithm for numerical integration of systems of ordinary differential equations (ODEs) with interval parameters and initial conditions. At each time moment, a piecewise polynomial function of a prescribed degree is constructed in the course of algorithm operation that interpolates the dependence of the solution on particular values of interval uncertainties with a controlled accuracy. We study the question of computational costs of the algorithm. An analytic estimate is derived for the number of operations; it depends on the algorithm parameters, in particular, the degree of interpolation and the specific features of the ODE system being integrated. Using a number of representative examples of various dimension and containing a varying number of interval uncertainties, we show that there exists an optimum value of interpolation degree from the viewpoint of computational costs.