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Exponential integrators

Published online by Cambridge University Press:  10 May 2010

Marlis Hochbruck  and
Alexander Ostermann
Marlis Hochbruck
Affiliation:
Karlsruher Institut für Technologie, Institut für Angewandte und Numerische Mathematik, D-76128 Karlsruhe, Germany, E-mail: marlis.hochbruck@kit.edu
Alexander Ostermann
Affiliation:
Institut für Mathematik, Universität Innsbruck, A-6020 Innsbruck, Austria, E-mail: alexander.ostermann@uibk.ac.at
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In this paper we consider the construction, analysis, implementation and application of exponential integrators. The focus will be on two types of stiff problems. The first one is characterized by a Jacobian that possesses eigenvalues with large negative real parts. Parabolic partial differential equations and their spatial discretization are typical examples. The second class consists of highly oscillatory problems with purely imaginary eigenvalues of large modulus. Apart from motivating the construction of exponential integrators for various classes of problems, our main intention in this article is to present the mathematics behind these methods. We will derive error bounds that are independent of stiffness or highest frequencies in the system.

Since the implementation of exponential integrators requires the evaluation of the product of a matrix function with a vector, we will briefly discuss some possible approaches as well. The paper concludes with some applications, in which exponential integrators are used.

Type
Research Article
Information
Acta Numerica , Volume 19 , May 2010 , pp. 209 - 286
Copyright
Copyright © Cambridge University Press 2010

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