Discretization Methods for Stable Initial Value Problems
- PDF / 10,709,499 Bytes
- 209 Pages / 468 x 684 pts Page_size
- 8 Downloads / 260 Views
1044 Eckart Gekeler
Discretization Methods for Stable Initial Value Problems
Spri nger-Verlag Berlin Heidelberg New York Tokyo 1984
Author
Eckart Gekeler Mathematisches Institut A der Universitat Stuttgart Pfaffenwaldring 57, 7000 Stuttgart 80, Federal Republic of Germany
AMS Subject Classifications (1980): 65L07, 65L20, 65M05, 65M10, 65M15,65M20 ISBN 3-540-12880-8 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-12880-8 Springer-Verlag New York Heidelberg Berlin Tokyo This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1984 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
Introduction
In the past twenty years finite element analysis has reached a high standard and also great progress has been achieved in the development of numerical procedures for stiff, i.e., stable and ill-conditioned differential systems since the communication of Dahlquist 1963. Both fields together provide the ingredients for a method of lines solution for partial differential equations. In this method time and space discretization are carried out independently of each other, which has the advantage that often available subroutine packages can be applied in one or both directions. Finite element or finite difference methods are used for the discretization in space direction and finite difference methods as multistep or Runge-Kutta methods are used for the numerical solution of the resulting semi-discrete system in time, as a rule. For example, if a hyperbolic initial boundary value problem with the differential equation
and suitable initial and boundary conditions is discretized by a finite element method or more generally by a Galerkin procedure then the semi-discrete system of ordinary differential equations has the form
(*)
My"
+
Ny'
+
Ky = c(t)
where M, N, and K are real symmetric and positive definite matrices. Mand N are wellconditioned but K is ill-conditioned in general, i.e, OKOOK- 10 » O. The finite element approximation of more general linear hyperbolic problems leads to similar systems. In engineering mechanics the basic partial differential equation is mostly not available because the body to be considered is too complex, instead the equations of motion are approximated by matrix structural analysis. The resulting 'equilibrium equations of dynamic finite element analysis' are then a large differential system for the displacements y being of the form (*) too. If also a number of eigenvalues of the associated generalized eigenvalue problem is wanted then methods employing eigenvector expansions may be preferred in the solution of (*) (modal analy
Data Loading...
