Explicit Symplectic Integrators Using Hessian--Vector Products

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Abstract
In 1991 Rowlands proposed an effectively fourth-order, effectively two-stage, explicit symplectic integrator based on using a Hessian-vector product to modify the force evaluation in the leapfrog method, and evidence indicates that for modest accuracy this method is highly competitive. Here we explore the possible existence of even more efficient fourth-order explicit symplectic integrators, also based on the use of Hessian-vector products and the concept of effective order. First it is shown that the cost of a force evaluation plus a Hessian-vector product is less than twice the cost of the force alone for a sum of two-body interactions. Then a new method is found that is generally better than both the method of Rowlands and that of Calvo, according to both a theoretical measure of the error and limited numerical experiments. The basic motivation behind the new method is quite simple: do a Hessian-vector computation only every other step, significantly cutting costs while only marginally increasing the error. The idea of effective order means that we allow for both the possibility of preprocessing the initial values before application of the basic method and the possibility of postprocessing the values obtained by the basic method, but at output points only. For some applications processing is unnecessary, but in any case processing has been shown to be possible at low additional cost. The derivation of the new method illustrates how to simplify by the use of II-series the determination of parameters for methods of increased effective accuracy.

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