An Introduction to Numerical Methods for ODEs and DAEs
Many standard implicit methods such as multistep and Runge-Kutta can be applied to both Ordinary Differential Equations (ODEs) and Differential Algebraic Equations (DAEs). This talk first compares and contrasts their ability to handle the two types of problems. Depending on the nature of the problem, methods that are effective for ODEs may break down for DAEs. Not all DAEs are solvable (that is, have a smooth solution). Even those that are solvable may have pathological behavior.
KeywordsDifferential Algebraic Equation Matrix Pencil Backward Differentiation Formula Perturbation Index Differential Index
Unable to display preview. Download preview PDF.
- Brenan, K. E., Stability and Convergence of Difference Approximations for Higher Index Differential-Algebraic Systems with Applications in Trajectory Control, PhD thesis, Department of Mathematics, UCLA, 1983Google Scholar
- Brenan, K. E., Campbell, S. L., and Petzold, L. R., The Numerical Solution of Initial Value Problems in Differential-Algebraic Equations, Elsevier Science Publishing Co, 1989Google Scholar
- Gantmacher, The Theory of Matrices, v2, Chelsea Publishing Co, New York, 1977Google Scholar
- Gear, C. W., Numerical Initial Values Problems in Ordinary Differential Equations, prentice Hah, 1971Google Scholar
- Gear, C. W. Differential Algebraic Equations, Indices, and Integral Algebraic Equations, Department of Computer Science Report 1505, University of Illinois at Urbana- Champaign, April, 1989, (submitted to SINUM).Google Scholar
- Gear, C. W., Keiper, G., The Analysis of Generalized BDF Methods Applied to Hessenberg Form DAEs, in preparation Google Scholar
- Hairer E., Lubich Ch., Roche M., The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods, Report, Universite de Geneve, Dept. de Mathematiques, Switzerland CH-1211, (1988).Google Scholar
- Keiper, G., Generalized BDF Methods Applied to Hessenberg Form DAES, PhD thesis, Department fo Computer Science, University of Illinois at Urbana-Champaign, 1989.Google Scholar
- Leimkuhler, B., Approximation Methods for the Consistent Initialization of Differential- Algebraic Equations, PhD thesis, Department of Computer Science Report 1450, University of Illinois at Urbana-Champaign, Aug, 1988.Google Scholar
- Leimkuhler, B., Somes Notes on Perturbations of Differential-Algebraic Equations, Institute of Mathematics Report A266, Helsinki University of Technology, May, 1989.Google Scholar
- Mrziglod, T., Zur Theorie und Numerischen Realisierung vob Losungsmethoden bei Differentialgleichungen mit Angekoppelten Algebraischen Gleichungen, PhD thesis, Universität zu Köln, 1987.Google Scholar
- Sincovec, R. F., Erisman, A. M., Yip, E. L., and Epton, M. A., Analysis of Descriptor Systems using Numerical Algorithms, IEEE Trans. Aut. Control AC-26, pp 139–147, (1981)Google Scholar