Parallel Störmer-Cowell methods for high-precision orbit computations

Many orbit problems in celestial mechanics are described by (nonstiff) initial-value problems (IVPs) for second-order ordinary differential equations of the form $y' = {bf f (y)$. The most successful integration methods are based on high-order Runge-Kutta-Nyström formulas. However, these methods were designed for sequential is paper, we consider high-order parallel methods that are not based on Runge-Kutta-Nyström formulas, but which fit into the class of general linear methods. In each step, these methods compute blocks of k approximate solution values (or stage values) at k different points using the whole previous block of solution values. The k stage values can be computed in parallel, so that on a k-processor computer system such methods effectively perform as a one-value method. The block methods considered in this paper are such that each equation defining a stage value resembles a linear multistep equation of the familiar Störmer-Cowell type. For k = 4 and k = 5 we constructed explicit PSC methods with stage order q = k and step point order p = k+1 and implicit PSC methods with q = k+1 and p = k+2. For k = 6 we can construct explicit 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 = 5 the abscissae of the stage values can be chosen such that only k-1 stage values 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 values needed, reduces to k* = k-1. The numerical examples reported in this paper show that the effective number of righthand 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-Nyström code DOPRIN (which is considered as one of the most efficient sequential codes for second-order ODEs). Furthermore, a comparison with the 12th-order parallel code PIRKN reveals that the PSC code is, in spite of its lower order, at least equally efficient, and in most cases more efficient than PIRKN.