Runge-Kutta time discretizations of nonlinear dissipative evolution equations

Author:
Eskil Hansen

Journal:
Math. Comp. **75** (2006), 631-640

MSC (2000):
Primary 65J15, 65M12

DOI:
https://doi.org/10.1090/S0025-5718-05-01866-1

Published electronically:
December 19, 2005

MathSciNet review:
2196983

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Global error bounds are derived for Runge-Kutta time discretizations of fully nonlinear evolution equations governed by -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 -convergence theory to infinite-dimensional spaces. An algebraically stable Runge-Kutta method with stage order is derived to have a global error which is at least of order or , 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

Email:
eskil@maths.lth.se

DOI:
https://doi.org/10.1090/S0025-5718-05-01866-1

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.