Abstract
Linear partial differential algebraic equations (PDAEs) of the form Au t(t, x) + Bu xx(t, x) + Cu(t, x) = f(t, x) are studied where at least one of the matrices A, B ∈ Rn×n is singular. For these systems we introduce a uniform differential time index and a differential space index. We show that in contrast to problems with regular matrices A and B the initial conditions and/or boundary conditions for problems with singular matrices A and B have to fulfill certain consistency conditions. Furthermore, two numerical methods for solving PDAEs are considered. In two theorems it is shown that there is a strong dependence of the order of convergence on these indexes. We present examples for the calculation of the order of convergence and give results of numerical calculations for several aspects encountered in the numerical solution of PDAEs.
Similar content being viewed by others
REFERENCES
M. Arnold, A note on the uniform perturbation index, Heft 52, Rostocker Math. Kolloquium, 1998.
M. Arnold and B. Simeon, The simulation of pantograph and catenary: A PDAE approach, Preprint Nr. 1990, Fachbereich Mathematik, TU Darmstadt, 1998.
K. E. Brenan, S. L. Campbell, and L. R. Petzold, Numerical solution of initial-value problems in differential-algebraic equations, North-Holland, Amsterdam, 1989.
S. L. Campbell and W. Marszalek, ODE/DAE integrators and MOL problems, ICIAM 95 Minisymposium on MOL, 1995.
S. L. Campbell and W. Marszalek, The index of an infinite dimensional implicit system, Math. Modelling Syst., 1:1 (1996), pp. 1–25.
S. L. Campbell and W. Marszalek, ODE/DAE integrators and MOL problems, ZAMM, 76:S1 (1996), pp. 251–254.
S. L. Campbell and W. Marszalek, DAEs arising from traveling wave solutions of PDEs, J. Comput. Appl. Math., 82:1–2 (1997), pp. 41–58.
C. W. Gear and L. R. Petzold, ODE methods for the solution of differential/algebraic systems, SIAM J. Numer. Anal., 21:4 (1984), pp. 716–728.
E. Griepentrog and R. März, Differential-algebraic Equations and their Numerical Treatment, Teubner-Texte zur Mathematik, Band 88, Leipzig, 1986.
P. Grindrod, The Theory and Applications of Reaction-diffusion Equations, Clarendon Press, Oxford, 1996.
E. Hairer, Ch. Lubich, and M. Roche, The Numerical Solution of Differential-algebraic Systems by Runge-Kutta Methods, Vol. 1409, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1989.
E. Hairer and G. Wanner, Solving Ordinary Differential Equations II, Springer-Verlag, Berlin, 1996.
A. W. Leung, Systems of Nonlinear Partial Differential Equations, Kluwer Academic Publishers, Dordrecht, 1989.
Ping Lin, A sequential regularization method for time-dependent incompressible Navier-Stokes equations, SIAM J. Numer. Anal., 34:3 (1997), pp. 1051–1071.
W. Lucht and K. Strehmel, Discretization based indices for semilinear partial differential algebraic equations, Appl. Numer. Math., 28 (1998), pp. 371–386.
W. Lucht, K. Strehmel, and C. Eichler-Liebenow, Linear partial differential algebraic equations, Part I: Indexes, consistent boundary/initial conditions, Report 17, Fachbereich Mathematik und Informatik, Martin-Luther-Universität, Halle, 1997.
W. Lucht, K. Strehmel, and C. Eichler-Liebenow, Linear partial differential algebraic equations, Part II: Numerical solution, Report 18, Martin-Luther-Universität Halle, Fachbereich Mathematik und Informatik, 1997.
W. Marszalek, Analysis of partial differential algebraic equations, PhD thesis, North Carolina State University, Raleigh, NC, 1997.
L. R. Petzold, Differential-algebraic equations are not ODE's, SIAM J. Sci. Stat. Comput., 3 (1982), pp. 367–384.
K. G. Pipilis, Higher order moving finite elements method for systems described by partial differential-algebraic equations, PhD thesis, Dept. of Chemical Engineering, Imperial College of Science, Technology and Medicine, London, 1990.
M. Sezgin, Magnetohydrodynamic flow in a rectangular channel, Internat. J. Numer. Methods Fluids, 7 (1987), pp. 697–718.
B. Simeon, Modelling a flexible slider crank mechanism by a mixed system of DAEs and PDEs, Math. Modelling Syst., 2(1):1–18, 1996.
G. Söderlind, Remarks on the stability of high-index DAEs with respect to parametric perturbations, Computing, 49 (1992), pp. 303–314.
J. W. Thomas, Numerical Partial Differential Equations: Finite Difference Methods, Springer-Verlag, New York, 1995.
W. Walter, Differential and Integral Inequalities, Springer-Verlag, New York, 1970.
J. Weickert, Navier-Stokes equations as a differential-algebraic system, Preprint SFB 393/96–08, Technische Universität Chemnitz-Zwickau, 1996.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Lucht, W., Strehmel, K. & Eichler-Liebenow, C. Indexes and Special Discretization Methods for Linear partial Differential Algebraic Equations. BIT Numerical Mathematics 39, 484–512 (1999). https://doi.org/10.1023/A:1022370703243
Issue Date:
DOI: https://doi.org/10.1023/A:1022370703243