EXPLICIT SYMPLECTIC INTEGRATORS USING HESSIAN-VECTOR PRODUCTS

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 ev idence indicates that for modest accuracy this method is highly compet itive. Here we explore the possible existence of even more efficient f ourth-order explicit symplectic integrators, also based on the use of Hessian-vector products and the concept of effective order. First it i s shown that the cost of a force evaluation plus a Hessian-vector prod uct is less than twice the cost of the force alone for a sum of two-bo dy interactions. Then a new method is found that is generally better t han both the method of Rowlands and that of Calvo, according to both a theoretical measure of the error and limited numerical experiments. T he basic motivation behind the new method is quite simple: do a Hessia n-vector computation only every other step, significantly cutting cost s while only marginally increasing the error. The idea of effective or der means that we allow for both the possibility of preprocessing the initial values before application of the basic method and the possibil ity of postprocessing the values obtained by the basic method, but at output points only. For some applications processing is unnecessary, b ut in any case processing has been shown to be possible at low additio nal cost. The derivation of the new method illustrates how to simplify by the use of II-series the determination of parameters for methods o f increased effective accuracy.