Solving ordinary differential equations on parallel computers — applied to dynamic rolling bearings simulation
In this paper we investigate how to solve certain kinds of ordinary differential equations (ODEs) efficiently on several types of MIMD parallel computers. The amount of parallelism for solving initial value problems such as ODEs is often quite limited, but by exploiting some characteristics of the application area where these problems are solved, the amount of parallelism can be increased. We focus on solving ODEs for rolling bearing dynamics simulation, which is computationally expensive. Typical characteristics of such ODEs are: stiff ODEs, very high numerical precision needed for solution, and computationally expensive to evaluate the derivatives.
We have adapted conventional solvers such as LSODA for execution on parallel computers, for example by evaluating the right-hand sides of the ODEs in parallel. The parallel machines used are: a Parsytec GigaCube with 16 T805 processors using the PARIX operating system, a Sun SPARCcenter 2000 with 8 processors and Solaris 2.3, and a cluster of nine SPARC 10 workstations connected via ethernet and using PVM. All these can be considered as Multiple Instruction Multiple Data (MIMD) architectures.
The obtained speedup was fairly good, approximately two thirds of linear speedup. However, this application requires rather fine-grained synchronization, which favours scheduling methods that minimize communication. As always, it is easier to get good speedups on machines with slower processors.
Unable to display preview. Download preview PDF.
- Iserles A. and Nörsett S. P. On the Theory of Parallel Runge-Kutta Methods. IMA Journal of numerical Analysis, 1990.Google Scholar
- P. K. Gupta. Advanced Dynamics of Rolling Elements. Springer-Verlag, New York, 1984.Google Scholar
- E. Hairer and G. Wanner. Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer-Verlag, 1991.Google Scholar
- Alan C. Hindmarsh. Brief Description of ODEPACK — A Systematized Collection of ODE Solvers. Computing & Mathematics Research Division, L-316 Lawrence Livermore National Laboratory Livermore, CA 94550, U.S.A., March 1987.Google Scholar
- Willard L. Miranker and Werner Liniger. Parallel Methods for the Numerical Integration of Ordinary Differential Equations. Kalle Petterson, 1966.Google Scholar
- P. Nordling. Efficient numerical algorithms for dynamic contact problems. Master's thesis, University of Linköping, 1993.Google Scholar
- Sommeijer. P. J, van ver Houwen B. P and Van Mourik P. A. Note on explicit parallel multistep Runge-Kutta methods. Journal of Computational and Applied Mathematics, 1989.Google Scholar
- B.P. Sommeijer, P. J. van der Houwen and W. Couzy. Embedded Diagonally Implicit Runge-Kutta Algorithms on Parallel Computers. Mathematic of Computation, 58(197): 135–159, January 1992.Google Scholar
- Freeman T. L and Phillips C. Parallel Numerical Algorithms. Prentice Hall International (UK) Ltd, 1992.Google Scholar
- P. J. van der Houwen and B. P. Sommeijer. Iterated Runge-Kutta Methods on Parallel Computers. SIAM J. SCI. STAT. COMPUT, 12(5): 1000–1028, September 1991.Google Scholar