NDLI: Explicit Exponential Runge--Kutta Methods for Semilinear Parabolic Problems

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Ostermann, Alexander Hochbruck, Marlis
Copyright Year	2005
Abstract	The aim of this paper is to analyze explicit exponential Runge--Kutta methods for the time integration of semilinear parabolic problems. The analysis is performed in an abstract Banach space framework of sectorial operators and locally Lipschitz continuous nonlinearities. We commence by giving a new and short derivation of the classical (nonstiff) order conditions for exponential Runge--Kutta methods, but the main interest of our paper lies in the stiff case. By expanding the errors of the numerical method in terms of the solution, we derive new order conditions that form the basis of our error bounds for parabolic problems. We show convergence for methods up to order four, and we analyze methods that were recently presented in the literature. These methods have classical order four, but they do not satisfy some of the new conditions. Therefore, an order reduction is expected. We present numerical experiments which show that this order reduction in fact arises in practical examples. Based on our new conditions, we finally construct methods that do not suffer from order reduction.
Starting Page	1069
Ending Page	1090
Page Count	22
File Format	PDF
ISSN	00361429
DOI	10.1137/040611434
e-ISSN	10957170
Issue Number	3
Volume Number	43
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-25
Access Restriction	Subscribed
Subject Keyword	order reduction Stability and convergence of numerical methods semilinear parabolic problems stiff order conditions exponential integrators Multistep, Runge-Kutta and extrapolation methods convergence Runge--Kutta methods explicit high-order methods
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Numerical Analysis Computational Mathematics

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