On solving nonlinear simultaneous equations arising from the double-exponential Sinc-collocation method for initial value problems

TL;DR

This paper analyzes the convergence of a fixed-point iteration method for solving nonlinear equations from the double-exponential Sinc-collocation method, demonstrating its efficiency and providing convergence conditions.

Contribution

It offers a convergence analysis and sufficient conditions for a Gauss-Seidel type iteration used in solving equations from the Sinc-collocation method.

Findings

The iteration typically reduces error by one or two orders per step.

A sufficient condition for global convergence is established.

Numerical examples confirm the theoretical analysis.

Abstract

The double-exponential Sinc-collocation method is known as a super-accurate method for solving initial value problems of ordinary differential equations, for which the error decreases almost exponentially as a function of the number of sample points in the temporal direction, $N$. However, this method requires solving nonlinear simultaneous equations in $nN$ variables when the problem dimension is $n$. Recently, Ogata pointed out that Gauss-Seidel type fixed-point iteration works surprisingly well for solving these equations, typically reducing the error by one or two orders of magnitude at each iteration. In this paper, we analyze the convergence of this iteration and give a sufficient condition for its global convergence. We also provide an upper bound on its convergence factor, which explains the efficiency of this iteration. Some numerical examples that illustrate the validity of our analysis are also provided.