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.
- Iserles A. and Nörsett S. P. On the Theory of Parallel Runge-Kutta Methods. IMA Journal of numerical Analysis, 1990.
- Gupta, P. K. (1984) Advanced Dynamics of Rolling Elements. Springer-Verlag, New York
- E. Hairer and G. Wanner. Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer-Verlag, 1991.
- Hindmarsh, A. C. (1987) 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.
- Willard L. Miranker and Werner Liniger. Parallel Methods for the Numerical Integration of Ordinary Differential Equations. Kalle Petterson, 1966.
- P. Nordling. Efficient numerical algorithms for dynamic contact problems. Master's thesis, University of Linköping, 1993.
- 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.
- Sommeijer, B.P., Houwen, P. J. v. d., Couzy, W. (1992) Embedded Diagonally Implicit Runge-Kutta Algorithms on Parallel Computers. Mathematic of Computation 58: pp. 135-159
- Freeman T. L and Phillips C. Parallel Numerical Algorithms. Prentice Hall International (UK) Ltd, 1992.
- Houwen, P. J. v. d., Sommeijer, B. P. (1991) Iterated Runge-Kutta Methods on Parallel Computers. SIAM J. SCI. STAT. COMPUT 12: pp. 1000-1028
- Solving ordinary differential equations on parallel computers — applied to dynamic rolling bearings simulation
- Book Title
- Parallel Scientific Computing
- Book Subtitle
- First International Workshop, PARA '94 Lyngby, Denmark, June 20–23, 1994 Proceedings
- pp 397-415
- Print ISBN
- Online ISBN
- Series Title
- Lecture Notes in Computer Science
- Series Volume
- Series ISSN
- Springer Berlin Heidelberg
- Copyright Holder
- Additional Links
- Industry Sectors
- eBook Packages
To view the rest of this content please follow the download PDF link above.