RKN-TYPE PARALLEL BLOCK PC METHODS WITH LAGRANGE-TYPE PREDICTORS

This paper describes the construction of block predictor-corrector met hods based on Runge-Kutta-Nystrom correctors. Our approach is to apply the predictor-corrector method not only at step points, but also at o ff-step points (block points), so that in each step, a whole block of approximations to the exact solution at off-step points is computed. I n the next step, these approximations are used to obtain a high-order predictor formula using Lagrange interpolation. By suitable choice of the abscissas of the off-step points, a much more accurately predicted value is obtained than by predictor formulas based on last step value s. Since the block of approximations at the off-step points can be com puted in parallel, the sequential costs of these block predictor-corre ctor methods are comparable with those of a conventional predictor-cor rector method. Furthermore, by using Runge-Kutta-Nystrom corrector met hods, the computation of the approximation at each off-step point is a lso highly parallel. Application of the resulting block predictor-corr ector methods to a few widely-used test problems reveals that the sequ ential costs are reduced by a factor ranging from 4 to 50 when compare d with the best sequential methods from the literature.