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Runge-Kutta time discretizations of nonlinear dissipative evolution equations

Author: Eskil Hansen
Journal: Math. Comp. 75 (2006), 631-640
MSC (2000): Primary 65J15, 65M12
Published electronically: December 19, 2005
MathSciNet review: 2196983
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Abstract: Global error bounds are derived for Runge-Kutta time discretizations of fully nonlinear evolution equations governed by $ m$-dissipative vector fields on Hilbert spaces. In contrast to earlier studies, the analysis presented here is not based on linearization procedures, but on the fully nonlinear framework of logarithmic Lipschitz constants in order to extend the classical $ B$-convergence theory to infinite-dimensional spaces. An algebraically stable Runge-Kutta method with stage order $ q$ is derived to have a global error which is at least of order $ q-1$ or $ q$, depending on the monotonicity properties of the method.

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Additional Information

Eskil Hansen
Affiliation: Centre for Mathematical Sciences, Lund University, Box 118, SE-221 00 Lund, Sweden

Keywords: Nonlinear evolution equations, logarithmic Lipschitz constants, $m$-dissipative maps, Runge-Kutta methods, algebraic stability, $B$-convergence
Received by editor(s): December 14, 2004
Published electronically: December 19, 2005
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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