Beiträge in Tagungsbänden:
W. Auzinger:
"Global error estimation in exponential integrators";
in: "Modelling and Stability",
herausgegeben von: Dynamical System Modelling and Stability Investigation;
DSMSI Dynamical System Modelling and Stability Investigation,
Kiew,
2013,
ISBN: 966-76-52-009,
S. 26
- 27.
Kurzfassung englisch:
GLOBAL ERROR ESTIMATION IN EXPONENTIAL INTEGRATORS
Auzinger W., Stolyarchuk R.R.
For the numerical approximation of the solution :[0, ] n u T of semilinear stiff ODE
systems
( u t) = Au(t) g(u(t)), u(0) given, (1)
exponential integrators are widely used. Here it is assumed that the linear part involving the stiff coefficient matrix n n A can be `exactly integrated', i.e., an efficient procedure for evaluation or accurate approximation of the mapping tA v e v is available. Exponential integrators of multistep type are based on reformulating (1) as a local integral equation via the variation-ofconstants
formula and approximate it by interpolatory quadrature. This leads to discrete schemes of the type 1 1 = ( ( ), , ( )) (stifforder hA
n n n n p n u e u hV g u g u p ), or (2)
1 1 1 = ( ( ), , ( ), ( )) (stifforder hA
n n n n p n n u e u hW g u g u g u p+1 ), (3) where h is the stepsize and ( ) = ( ) n n u u t u nh . These schemes are generalizations of the classical (explicit) Adams-Bashforth and (implicit) Adams-Moulton schemes. The terms ( ) n hV and ( ) n hW are multistep approximations of the variation-of-constants integral over the interval 1 [ , ] n n t t which involve further evaluations of exponentials.
The numerical realization of (3) is much more involved than for (2) because each step involves the solution of a nonlinear system. As in the classical (non-stiff) multistep context, there are various ways to combine (2) with (3), e.g., in a predictor-corrector type fashion. Here we consider another option, namely a procedure for estimating the global error 1 1 ( ) n n u u t of (2) in an a posteriori sense, by means of . computing the defect (residual) of n 1 u with respect to (3) in each step, and . backsolving for a global error estimate by an integration involving the defect, using a simple auxiliary scheme like exponential Euler.
This way of estimating the error is called defect integration. Here, n u and its error estimate are determined simultaneously.
We present relevant details of this error estimation procedure and demonstrate its effectiveness for a nonlinear stiff test problem. We also include results for rational integrators, where hA e is approximated by an A-stable p -th order Padé approximation. Furthermore we indicate how to extend the existing convergence theory for (2) in order to prove the asymptotical correctness of the estimator (work in progress).
Elektronische Version der Publikation:
http://www.dsmsi.univ.kiev.ua/downloads/book_DSMSI-2013.pdf
Erstellt aus der Publikationsdatenbank der Technischen Universität Wien.