Solving implicit differential equations on parallel computers

We construct and analyse three methods for solving initial value problems for implicit differential equations (IDEs) on parallel computer systems. The first IDE method can be applied to general IDEs of higher index, the other two methods can be applied to partitioned (or semi-explicit) IDEs. The partitioned IDE methods both exploit the special form of the problem and often converge faster than the general IDE method. The first partitioned IDE method is suitable for higher-index problems, the second partitioned IDE method only applies to index 1 problems, but possesses more parallelism across the method. The convergence of these methods is illustrated by solving implicit IDEs of index 0 until 3 that are taken from the literature

Step-parallel algorithms for stiff initial value problems

For the parallel integration of stiff initial value problems, three types of parallelism can be employed: 'parallelism across the problem', 'parallelism across the method' and 'parallelism across the steps'. Recently, methods based on Runge-Kutta schemes that use parallelism across the method have been proposed in [5, 6]. These methods solve implicit Runge-Kutta schemes by means of the so-called diagonally iteration scheme and are called PDIRK methods. The experiments described in [5], show that the speedup factor of certain high-order PDIRK methods, is about 2 with respect to a good sequential code. However, a disadvantage of the high-order PDIRK methods is, that a relatively large number of iterations is needed for each step. This disadvantage can be compensated by employing step-parallelism. Step-parallel methods are methods in which a number of steps are treated simultaneously. This form of parallelism can be applied to any predictor-corrector method. A common feature of this approach is their poor convergence behaviour, unless the various strategies are carefully designed. In the present paper, we describe two strategies for the PDIRK across the steps method. Example problems tested in this paper show for the best strategy, a speed-up factor ranging from 4 to 7 with respect to the best sequential codes

Parallel iteration across the steps of high-order Runge-Kutta methods for nonstiff initial value problems

AbstractFor the parallel integration of nonstiff initial value problems (IVPs), three main approaches can be distinguished: approaches based on “parallelism across the problem”, on “parallelism across the method” and on “parallelism across the steps”. The first type of parallelism does not require special integration methods and can be exploited within any available IVP solver. The method-parallelism approach received much attention, particularly within the class of explicit Runge-Kutta methods originating from fixed point iteration of implicit Runge-Kutta methods of Gaussian type. The construction and implementation on a parallel machine of such methods is extremely simple. Since the computational work per processor is modest with respect to the number of data to be exchanged between the various processors, this type of parallelism is most suitable for shared memory systems. The required number of processors is roughly half the order of the generating Runge-Kutta method and the speed-up with respect to a good sequential IVP solver is about a factor 2. The third type of parallelism (step-parallelism) can be achieved in any IVP solver based on predictor-corrector iteration and requires the processors to communicate after each full iteration. If the iterations have sufficient computational volume, then the step-parallel approach may be suitable for implementation on distributed memory systems. Most step-parallel methods proposed so far employ a large number of processors, but lack the property of robustness, due to a poor convergence behaviour in the iteration process. Hence, the effective speed-up is rather poor. The dynamic step-parallel iteration process proposed in the present paper is less massively parallel, but turns out to be sufficiently robust to achieve speed-up factors up to 15

PSIDE users' guide

PSIDE -- Parallel Software for Implicit Differential Equations -- is a code for solving implicit differential equations on shared memory parallel computers. In this paper we describe the user interface

Specification of PSIDE

PSIDE is a code for solving implicit differential equations on parallel computers. It is an implementation of the four-stage Radau IIA method. The nonlinear systems are solved by a modified Newton process, in which every Newton iterate itself is computed by an iteration process. This process is constructed such that the four stage values can be computed simultaneously. We describe here how PSIDE is set up as a modular system and what control strategies have been chosen

Parallel iterative linear solvers for multistep Runge-Kutta methods

This paper deals with solving stiff systems of differential equations by implicit Multistep Runge--Kutta (MRK) methods. For this type of methods, nonlinear systems of dimension sd arise, where s is the number of Runge--Kutta stages and d the dimension of the problem. Applying a Newton process leads to linear systems of the same dimension, which can be very expensive to solve in practice. Like in [HS96], where the one-step RK methods were considered, we approximate these linear systems by s systems of dimension d, which can be solved in parallel on a computer with s processors. In terms of Jacobian evaluations and LU-decompositions, the k-step s-stage MRK applied with PILSMRK on s processors is equally expensive as the widely used k-step Backward Differentiation Formula on 1 processor, whereas the stability properties are better than that of BDF. If both methods perform the same number of Newton iterations, then the accuracy delivered by the new method is also higher than that of BDF

Test set for IVP solvers

In this paper a collection of Initial Value test Problems for systems of Ordinary Differential Equations, Implicit Differential Equations and Differential-Algebraic Equations is presented. This test set is maintained by the project group for Parallel IVP Solvers of CWI, department of Numerical Mathematics. This group invites everyone to contribute new test problems to this test set. How new problems can be submitted can be found in this paper as well