P.J. van der Houwen

Parallel iteration of high-order Runge-Kutta methods with stepsize control

Abstract This paper investigates iterated Runge-Kutta methods of high order designed in such a way that the right-hand side evaluations can be computed in parallel. Using stepsize control based on embedded formulas a highly efficient code is developed. On parallel computers, the 8th-order mode of this code is more efficient than the DOPR18 implementation of the formulas of Prince and Dormand. The 10th-order mode is about twice as cheap for comparable accuracies.

On the Internal Stability of Explicit, m‐Stage Runge‐Kutta Methods for Large m‐Values

Explicit, m-stage Runge-Kutta methods are derived for which the maximal stable integration step per right hand side evaluation is proportional to m when applied to semi-discrete parabolic initial-boundary value problems. The internal stability behaviour of these methods is compared with that of similar Runge-Kutta methods proposed in the literature. Both by analysis and by numerical experiments we show that the value of m in the schemes proposed in this paper is not restricted by internal instabilities.

Iterated Runge-Kutta methods on parallel computers

This paper examines diagonally implicit iteration methods for solving implicit Runge–Kutta methods with high stage order on parallel computers. These iteration methods are such that after a finite number of m iterations, the iterated Runge–Kutta method belongs to the class of diagonally implicit Runge–Kutta methods (DIRK methods) using

Stability in linear multistep methods for pure delay equations

mk

Parallel Adams methods

implicit stages where k is the number of stages of the generating implicit Runge–Kutta method (corrector method). However, a large number of the stages of this DIRK method can be computed in parallel, so that the number of stages that have to be computed sequentially is only m. The iteration parameters of the method are tuned in such a way that fast convergence to the stability characteristics of the corrector method is achieved. By means of numerical experiments it is also shown that the solution produced by the resulting iteration method converges rapidly to the corrector solution so that both stability and accuracy characteristics are comparable with those of the corrector. This implies that the reduced accuracy often shown when integrating stiff problems by means of DIRK methods already available in the literature (which is caused by a low stage order) is not shown by the DIRK methods developed in this paper, provided that the corrector method has a sufficiently high stage order.

Parallel block predictor-corrector methods of Runge-Kutta type

Abstract The stability regions of linear multistep methods for pure delay equations are compared with the stability region of the delay equation itself. A criterion is derived stating when the numerical stability region contains the analytical stability region. This criterion yields an upper bound for the integration step (conditional Q-stability). These bounds are computed for the Adams-Bashforth, Adams-Moulton and backward differentiation methods of orders ⩽8. Furthermore, symmetric Adams methods are considered which are shown to be unconditionally Q-stable. Finally, the extended backward differentiation methods of Cash are analysed.

A-stable parallel block methods for ordinary and integro-differential equations

Abstract In the literature, various types of parallel methods for integrating nonstiff initial-value problems for first-order ordinary differential equation have been proposed. The greater part of them are based on an implicit multistage method in which the implicit relations are solved by the predictor-corrector (or fixed point iteration) method. In the predictor-corrector approach the computation of the components of the stage vector iterate can be distributed over s processors, where s is the number of implicit stages of the corrector method. However, the fact that after each iteration the processors have to exchange their just computed results is often mentioned as a drawback, because it implies frequent communication between the processors. Particularly on distributed memory computers, such a fine grain parallelism is not attractive. An alternative approach is based on implicit multistage methods which are such that the implicit stages are already parallel, so that they can be solved independently of each other. This means that only after completion of a step, the processors need to exchange their results. The purpose of this paper is the design of a class of parallel methods for solving nonstiff IVPs. We shall construct explicit methods of order k + 1 with k parallel stages where each stage equation is of Adams-Bashforth type and implicit methods of order k + 2 with k parallel stages which are of Adams-Moulton type. The abscissae in both families of methods are proved to be the Lobatto points, so that the Adams-Bashforth type method can be used as a predictor for the Adams-Moulton-type corrector.

Analysis of parallel diagonally implicit iteration of Runge-Kutta methods

Abstract 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 proceding 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.

Analysis of approximate factorization in iteration methods

Abstract In this paper we study the stability of a class of block methods which are suitable for integrating ordinary and integro-differential equations on parallel computers. A-stable methods of orders 3 and 4 and A(α)-stable methods with α > 89.9° of order 5 are constructed. On multiprocessor computers these methods are of the same computational complexity as implicit linear multistep methods on one-processor computers.

Symmetric linear multistep methods for second-order differential equations with periodic solutions

Abstract In this paper, we analyze parallel, diagonally implicit iteration of Runge-Kutta methods (PDIRK methods) for solving large systems of stiff equations on parallel computers. Like Newton-iterated backward differentiation formulas (BDFs), these PDIRK methods are such that in each step the (sequential) costs consist of solving a number of linear systems with the same matrix of coefficients and with the same dimension as the system of differential equations. Although for PDIRK methods the number of linear systems is usually higher than for Newton iteration of BDFs, the more computationally intensive work of computing the matrix of coefficients and its LU-decomposition are identical. The advantage of PDIRK methods over Newton-iterated BDFs is their unconditional stability (A-stability for Gauss-based methods and L-stability for Radau-based methods) for any order of accuracy. Special characteristics of the PDIRK methods will be studied, such as the rate of convergence, the influence of particular predictors on the resulting stability properties, and the stiff error constants in the global error.