Adaptive Interpolation Algorithm Based on a kd-Tree for Numerical Integration of Systems of Ordinary Differential Equations with Interval Initial Conditions

Abstract

We consider issues related to the numerical solution of interval systems of ordinary differential equations. We suggest an algorithm that permits finding interval estimates of solutions with prescribed accuracy in reasonable time. The algorithm constructs an adaptive partition (a dynamic structured grid) based on a kd-tree over the space formed by interval initial conditions for the ordinary differential equations. In the operation of the algorithm, a piecewise polynomial function interpolating the dependence of the solution on the specific values of interval parameters is constructed at each step of solution of the original problem. We prove that the global error estimate linearly depends on the height of the kd-tree. The algorithm is tested on several examples; the test results show its efficiency when solving problems of the class under study.