Error Analysis of Symplectic Multiple Time Stepping

Abstract

Most time-dependent problems obtain only limited benefit from variable stepsize, and to exploit more fully the existence of multiple time scales, it is necessary to use a multiplicity of stepsizes simultaneously. For example, many systems of ODEs have right-hand sides which naturally decompose into a sum of terms with different time scales, in which case the various terms can be sampled with different stepsizes-an idea called multiple time stepping (MTS). Addressed here is the question of how to choose these stepsizes in the context of molecular dynamics. This application and others are often modeled by Hamiltonian systems, for which symplectic integrators are often preferred because they introduce discretization error that can be very nearly interpreted as a perturbation to the scalar Hamiltonian function itself. The method analyzed here is a symplectic MTS method that has received recent attention. Analyzed is a problem that models the bonded interactions in molecular dynamics (MD). The method of analysis is quite novel in that it utilizes a slight change of variables that is more favorable to the numerical solution without seriously affecting the quantities of interest. The study concludes with a specific prescription for determining stepsize ratios.