Tuning Symplectic Integrators is Easy and Worthwhile

Abstract

Many applications in computational physics that use numerical integrators based on splitting and composition can benefit from the development of optimized algorithms and from choosing the best ordering of terms. The cost in programming and execution time is minimal, while the performance improvements can be large. In this note we report the influence of term ordering for random systems and for two systems from celestial mechanics that describe particle paths near black holes, quantifying its significance for both optimized and unoptimized methods. We also present a method for the computation of solutions of integrable monomial Hamiltonians that minimizes roundoff error and allows the effective use of compensation summation.