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Zbl 1242.65109
Hochbruck, Marlis; Ostermann, Alexander
Exponential integrators. (English)
[J] Acta Numerica 19, 209-286 (2010). ISSN 0962-4929; ISSN 1474-0508/e

Summary: 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 islo present the mathematics behind these methods. We will derive error bounds that are independent of stiffness or highest frequencies in the system. \par 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.

MSC 2000:: *65J08
65L05 Initial value problems for ODE (numerical methods)
34G20 Nonlinear ODE in abstract spaces
35K55 Nonlinear parabolic equations

Keywords: exponential integrators; stiff problems; parabolic partial differential equations; highly oscillatory problems; error bounds

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Zentralblatt MATH Berlin [Germany]