62 research outputs found

    Parallel Adams methods

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    AbstractIn 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

    Applying Parallel Processing Techniques to Tether Dynamics Simulation

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    The focus of this research has been to determine the effectiveness of applying parallel processing techniques to a sizable real-world problem, the simulation of the dynamics associated with a tether which connects two objects in low earth orbit, and to explore the degree to which the parallelization process can be automated through the creation of new software tools. The goal has been to utilize this specific application problem as a base to develop more generally applicable techniques

    Parallel step-by-step methods

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    Group implicit concurrent algorithms in nonlinear structural dynamics

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    During the 70's and 80's, considerable effort was devoted to developing efficient and reliable time stepping procedures for transient structural analysis. Mathematically, the equations governing this type of problems are generally stiff, i.e., they exhibit a wide spectrum in the linear range. The algorithms best suited to this type of applications are those which accurately integrate the low frequency content of the response without necessitating the resolution of the high frequency modes. This means that the algorithms must be unconditionally stable, which in turn rules out explicit integration. The most exciting possibility in the algorithms development area in recent years has been the advent of parallel computers with multiprocessing capabilities. So, this work is mainly concerned with the development of parallel algorithms in the area of structural dynamics. A primary objective is to devise unconditionally stable and accurate time stepping procedures which lend themselves to an efficient implementation in concurrent machines. Some features of the new computer architecture are summarized. A brief survey of current efforts in the area is presented. A new class of concurrent procedures, or Group Implicit algorithms is introduced and analyzed. The numerical simulation shows that GI algorithms hold considerable promise for application in coarse grain as well as medium grain parallel computers

    Stability control for approximate implicit time-stepping schemes with minimal residual iterations

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    Implicit schemes for the integration of ODE's are popular when stabil- ity is more of concern than accuracy, for instance for the computation of a steady state solution. However, in particular for very large sys- tems the solution of the involved linear systems maybevery expensive. In this paper we study the solution of these linear systems by a mod- erate number of iterations of the minimum residual iterative method GMRES. Of course, this puts limits to the step size since these ap- proximate schemes may be viewed as explicit schemes and these are never unconditionally stable. It turns out that even a modest degree of approximationallows rather large time steps and we propose a simple mechanism for the control of the step size with respect to stability

    Block Runge-Kutta methods

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    Continuous variable stepsize explicit pseudo two-step RK methods

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    AbstractThe aim of this paper is to apply a class of constant stepsize explicit pseudo two-step Runge-Kutta methods of arbitrarily high order to nonstiff problems for systems of first-order differential equations with variable stepsize strategy. Embedded formulas are provided for giving a cheap error estimate used in stepsize control. Continuous approximation formulas are also considered for use in an eventual implementation of the methods with dense output. By a few widely used test problems, we compare the efficiency of two pseudo two-step Runge-Kutta methods of orders 5 and 8 with the codes DOPRI5, DOP853 and PIRK8. This comparison shows that in terms of ƒ-evaluations on a parallel computer, these two pseudo two-step Runge-Kutta methods are a factor ranging from 3 to 8 cheaper than DOPRI5, DOP853 and PIRK8. Even in a sequential implementation mode, fifth-order new method beats DOPRI5 by a factor more than 1.5 with stringent error tolerances
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