ODE solvers using band-limited approximations

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 ϵ, so that they integrate functions eibx, for all |b|⩽c, with accuracy ϵ. 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), for orbit computations in astrodynamics. Since BLC-IRK minimizes the number of nodes needed to obtain the solution, in this problem we achieve speed close to that of the traditional explicit multistep methods.