VarInt

Abstract

Geometric numerical integration refers to a class of numerical integration algorithms that preserve the differential geometric structure that defines the evolution of dynamical systems. For simulations over relatively short time spans, as compared to the intrinsic time scales, standard (non-geometric) integrators are often advantageous, as they include adaptive and multistep methods, which can be both accurate and fast. Extended computations require a different, geometric approach, as non-geometric methods tend to generate or dissipate energy artificially due to the fact that they do not respect the fundamental geometry of the phase flow, which means that at some point the errors dominate. It is common to create geometric numerical integrators based on either previous knowledge of classes of non-geometric numerical integrators, from which specific instances can be derived that are geometric, or truncated series solutions to the Hamilton--Jacobi equation for (canonical) transformations near the identity. Unfortunately, the actual calculations for higher-order geometric numerical algorithms are generally quite involved. There is, however, an alternative strategy that avoids many of the difficulties inherent in the design of higher-order versions of these geometric numerical integrators. It relies on the discretization of the action functional, from which one derives the numerical algorithms in a straightforward manner. These so-called variational integrators preserve the differential geometric structure automatically; all conserved quantities are preserved infinitesimally, too. An easy-to-use and freely available package named VarInt is presented that enables one to generate and analyse new variational integrators systematically to arbitrary order with Maple. All VarInt requires from the user is the action and a quadrature formula to approximate it. One can either select one of the built-in quadrature rules, or one supplies it manually. With VarInt it is now possible to venture beyond the standard geometric numerical integrators, without the need for an advanced appreciation of the mathematical details. Several numerical examples, obtained with a basic numerical analysis tool in VarInt, demonstrate the superior performance of new variational integrators for certain classes of dynamical systems. VarInt is ideally suited for researchers and engineers who wish to design, study, test and/or analyse new geometric numerical integration algorithms without the hassle of laborious computations. All algorithms can be tuned to the required level of specificness of the problem at hand due to Maples symbolic capabilities and its code optimization procedures. In addition, VarInt can be of value in the classroom, as a tool to assist in increasing the understanding of variational integrators.