ODE SOLVERS USING BAND-LIMITED APPROXIMATIONS

Abstract. We use generalized Gaussian quadratures for expo-nentials to develop a new ODE solver. Nodes and weights of these quadratures are computed for a given bandlimit c and user selected accuracy ǫ, so that they integrate functions eibx, for all |b | ≤ c, with accuracy ǫ. Nodes of these quadratures do not concentrate exces-sively 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 (numeri-cally) 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. 1