ODE Solvers Using Bandlimited Approximations

TL;DR

This paper introduces a novel ODE solver based on bandlimited approximations and generalized Gaussian quadratures, enabling efficient, stable, and accurate solutions for functions with bounded frequency content, demonstrated in orbit determination.

Contribution

The paper develops a new ODE solver using bandlimited quadratures that avoids endpoint node concentration, allowing large node counts and demonstrating symplecticity and A-stability.

Findings

The solver is symplectic and numerically A-stable.

It achieves near-explicit method speeds in orbit determination.

Nodes do not concentrate near interval endpoints, unlike polynomial Gaussian quadratures.

Abstract

We use generalized Gaussian quadratures for exponentials to develop a new ODE solver. Nodes and weights of these quadratures are computed for a given bandlimit $c$ and user selected accuracy $\epsilon$, so that they integrate functions $e^{ibx}$, for all $|b|\le c$, with accuracy $\epsilon$. Nodes of these quadratures do not concentrate excessively near the end points of an interval as those of the standard, polynomial-based Gaussian quadratures. Due to this property, the usual implicit Runge Kutta (IRK) collocation method may be used with a large number of nodes, as long as the method chosen for solving the nonlinear system of equations converges. We show that the resulting ODE solver is symplectic and demonstrate (numerically) that it is A-stable. We use this solver, dubbed Band-limited Collocation (BLC-IRK), in the problem of orbit determination. Since BLC-IRK minimizes the number of nodes needed to obtain the solution, in this problem we achieve speed close to that of explicit multistep methods.