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W. Auzinger:
"Global error estimation in exponential integrators";
in: "Modelling and Stability", issued by: Dynamical System Modelling and Stability Investigation; DSMSI Dynamical System Modelling and Stability Investigation, Kiew, 2013, ISBN: 966-76-52-009, 26 - 27.

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).

http://www.dsmsi.univ.kiev.ua/downloads/book_DSMSI-2013.pdf