Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 1093.65052
Hochbruck, Marlis; Ostermann, Alexander
Explicit exponential Runge-Kutta methods for semilinear parabolic problems. (English)
[J] SIAM J. Numer. Anal. 43, No. 3, 1069-1090 (2005). ISSN 0036-1429; ISSN 1095-7170/e

Summary: 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.

MSC 2000:: *65J15 Equations with nonlinear operators (numerical methods)
65M12 Stability and convergence of numerical methods (IVP of PDE)
65L06 Multistep, Runge-Kutta, and extrapolation methods
34G20 Nonlinear ODE in abstract spaces
65L70 Error bounds (numerical methods for ODE)
35K55 Nonlinear parabolic equations
65M06 Finite difference methods (IVP of PDE)
65M15 Error bounds (IVP of PDE)

Keywords: exponential integrators; Runge--Kutta methods; semilinear parabolic problems; stiff order conditions; explicit high-order methods; convergence; order reduction; Banach space; error bounds; numerical experiments

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]