Adaptive Piecewise Poly-Sinc Methods for Ordinary Differential Equations

TL;DR

This paper introduces an adaptive piecewise Poly-Sinc collocation method for solving ordinary differential equations, achieving exponential convergence and validated on both regular and stiff equations.

Contribution

It extends Poly-Sinc approximation to adaptive ODE solving with proven exponential convergence and a statistical partition refinement approach.

Findings

Achieves exponential convergence in iterations.

Validates effectiveness on regular and stiff ODEs.

Provides an a priori error estimate for the method.

Abstract

We propose a new method of adaptive piecewise approximation based on Sinc points for ordinary differential equations. The adaptive method is a piecewise collocation method which utilizes Poly-Sinc interpolation to reach a preset level of accuracy for the approximation. Our work extends the adaptive piecewise Poly-Sinc method to function approximation, for which we derived an a priori error estimate for our adaptive method and showed its exponential convergence in the number of iterations. In this work, we show the exponential convergence in the number of iterations of the a priori error estimate obtained from the piecewise collocation method, provided that a good estimate of the exact solution of the ordinary differential equation at the Sinc points exists. We use a statistical approach for partition refinement. The adaptive greedy piecewise Poly-Sinc algorithm is validated on regular and stiff ordinary differential equations.