Parallel defect control

Abstract

How can small-scale parallelism best be exploited in the solution of nonstiff initial value problems? It is generally accepted that only modest gains inefficiency are possible, and it is often the case that ldquofastrdquo parallel algorithms have quite crude error control and stepsize selection components. In this paper we consider the possibility of using parallelism to improvereliability andfunctionality rather than efficiency. We present an algorithm that can be used with any explicit Runge-Kutta formula. The basic idea is to take several smaller substeps in parallel with the main step. The substeps provide an interpolation facility that is essentially free, and the error control strategy can then be based on a defect (residual) sample. If the number of processors exceeds (p - 1)/2, wherep is the order of the Runge-Kutta formula, then the interpolant and the error control scheme satisfy very strong reliability conditions. Further, for a given orderp, the asymptotically optimal values for the substep lengths are independent of the problem and formula and hence can be computed a priori. Theoretical comparisons between the parallel algorithm and optimal sequential algorithms at various orders are given. We also report on numerical tests of the reliability and efficiency of the new algorithm, and give some parallel timing statistics from a 4-processor machine

Block Methods with Off-Steps Points for Solving First Order Ordinary Differential Equations

“…To obtain the coefficient γ m , we apply the Maclaurin Series in (4). Recall that the Maclaurin Series is the Taylor Series expansion of a function about 0.…”

Parallel block predictor-corrector methods of Runge-Kutta type

“…Recalling that the 15 digits experiments reported in [7] indicate that a minimal spacing of 0.2 is acceptable in the case of Hermite interpolation, we expect that on 15-digit computers and for orders up to p = 10, an averaged spacing of 2/(p + 2) should be acceptable in the case of the more stable Lagrange interpolation formulas. We remark that the optimal location of the off-step points for defect control as derived in [7] is in the interval where the defect is to be computed, rather than advancing the current step point as in (2.12).…”

“…Probably, the most realistic option is a limitation on the minimal spacing of the abscissas ai. In [7] where Hermite interpolation formulas were used for deriving reliable error estimates for defect control, it was found that on a Silicon Grafics Inc. Power Iris 4D /240S-64 machine with 15 digits precision, the abscissas should be separated by 0.2 in order to suppress rounding errors. For the more stable Lagrange interpolation formulas, we expect that slightly smaller spacings are still acceptable.…”

“…We remark that the optimal location of the off-step points for defect control as derived in [7] is in the interval where the defect is to be computed, rather than advancing the current step point as in (2.12). Finally, we remark that the abscissas defined by (2.12) enable us to develop various cheap strategies for stepsize control.…”

“…A further increase of the amount of parallelism in step-by-step methods consists of computing parallel solution values not only at step points, but also at off-step points, so that, in each step, a whole block of approximations to the exact solution is computed. This approach was successfully used in [7] for obtaining reliable defect control in explicit RK methods. In this paper, we want to use this approach for constructing parallel PC methods where the value of k is substantially less than the order p and where, at the same time, the effective stability boundaries are acceptably large.…”