Parallel block predictor-corrector methods of Runge-Kutta type

Abstract

Van der Houwen, P.J. and Nguyen huu Cong, Parallel block predictor-corrector methods of Runge-Kutta type, Applied Numerical Mathematics 13 (1993) 109-123.In this paper, we construct block predictor-corrector methods using Runge-Kutta correctors. Our approach consists of applying the predictor-corrector method not only at step points, but also at off-step points (block points), so that, in each step, a whole block of approximations to the exact solution is computed. In the next step, these approximations are used to obtain a high-order predictor formula by Lagrange or Hermite interpolation. By choosing the abscissas of the off-step points narrowly spaced, a much more accurately predicted value is obtained than by predictor formulas based on preceding step point values. Since the approximations at the off-step points to be computed in each step can be obtained in parallel, the sequential costs of these block predictor-corrector methods are comparable with those of a conventional predictor-corrector method. Furthermore, by using Runge-Kutta correctors, the predictor-corrector iteration scheme itself is also highly parallel. Application of these block predictor-corrector methods based on Lagrange-Gauss pairs to a few widely-used test problems reveals that the sequential costs are reduced by a factor ranging from 2 to 11 when compared with the best sequential methods.

Explicit, high-order Runge-Kutta-Nyström methods for parallel computers

“…In this way, r approximations are found in each step and this information is used to create a high-order prediction in the next step. Compared with the original PIRK method (i.e., with blocklength r = 1), a substantial increase in efficiency is reported in [17]. This idea is of course equally well applicable in the present context of second-order equations.…”

“…The information needed to construct such a prediction could be obtained from approximations calculated in the preceding step. For example, in [17], van der Houwen and Nguyen huu Cong analyse (for first-order ODEs) a block version of the PIRK method [15] with the aim to construct a high-order prediction. The idea is to apply a PIRK method (which is very similar to a PIRKN method) not only with stepsize h, but, in addition (and simultaneously) with stepsizes h; = a;h, i = 1, ... , r -1.…”

Collocation-based two-step Runge-Kutta methods

“…However, since the matrix Q is a Vandermonde one, the closeness of the abscissae will increase the magnitude of the entries of the matrix (Bs-k,8-k, B~-k,k, v), causing serious round-off errors in the approximated values. There are several ways to reduce this round-off effect (see Subsection 2.3 in [10]) but the most realistic option is to limit the minimal spacing of the abscissae of collocation vector c. It was found that on a Silicon Grafics Inc. Power Iris 4D/240S-64 machine with 15 digits precision, the abscissae should be separated by about 0.2 in order to suppress rotmding errors as muchas possible (cf. also [10]).…”

Improved parallel-iterated pseudo two-step RK methods for nonstiff IVPs

“…The rate of convergence of the IPIPTRK method (3.1) is defined by using the model test equation y (t) = λy(t), where λ runs through the eigenvalues of the Jacobian matrix ∂f/∂y (cf., e.g., [7,11,14,20]). Applying the IPIPTRK method (3.1) to this model test equation, we obtain the iteration error equation…”