Parallel Stormer-Cowell methods for high-precision orbit computations

Many orbit problems in celestial mechanics are described by (nonstiff) init ial-value problems (IVPs) for second-order ordinary differential equations of the form y" = f(y). The most successful integration methods are based on high-order Runge-Kutta-Nystrom formulas. However, these methods were desig ned for sequential computer systems. In this paper, we consider high-order parallel methods that are not based on Runge-Kutta-Nystrom formulas, but wh ich fit into the class of general linear methods. In each step, these metho ds compute blocks of k approximate solution values (or stage values) at k d ifferent points using the whole previous block of solution values. The k st age values can be computed in parallel, so that on a k-processor computer s ystem such methods effectively perform as a one-value method. The block met hods considered in this paper are such that each equation defining a stage value resembles a linear multistep equation of the familiar Stormer-Cowell type. For k = 4 and k = 5 we constructed explicit PSC methods with stage or der q = k and step point order p = k + 1 and implicit PSC methods with q = k + 1 and p = k + 2. For k greater than or equal to 6 we can construct expl icit PSC methods with q = k and p = k + 2 and implicit PSC methods with q = k + 1 and p = k + 3. It turns out that for k greater than or equal to 5 th e abscissae of the stage values can be chosen such that only k - 1 stage va lues in each block have to be computed, so that the number of computational stages, and hence the number of processors and the number of starting valu es needed, reduces to k* = k - 1. The numerical examples reported in this p aper show that the effective number of right-hand side evaluations required by a variable stepsize implementation of the 10th-order PSC method is 4 up to 30 times less than required by the Runge-Kutta-Nystrom code DOPRIN (whi ch is considered as one of the most efficient sequential codes for second-o rder ODEs). Furthermore, a comparison with the 12th-order parallel code PIR KN reveals that the PSC code is, in spite of its lower order, at least equa lly efficient, and in most cases more efficient than PIRKN. (C) 1999 Elsevi er Science B.V. and IMACS. All rights reserved.