Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 

 

Algorithms without accuracy saturation for evolution equations in Hilbert and Banach spaces


Authors: Ivan P. Gavrilyuk and Volodymyr L. Makarov
Journal: Math. Comp. 74 (2005), 555-583
MSC (2000): Primary 65J10, 65M70; Secondary 35K90, 35L90
DOI: https://doi.org/10.1090/S0025-5718-04-01720-X
Published electronically: October 27, 2004
MathSciNet review: 2114638
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the Cauchy problem for the first and the second order differential equations in Banach and Hilbert spaces with an operator coefficient $A(t)$ depending on the parameter $t$. We develop discretization methods with high parallelism level and without accuracy saturation; i.e., the accuracy adapts automatically to the smoothness of the solution. For analytical solutions the rate of convergence is exponential. These results can be viewed as a development of parallel approximations of the operator exponential $e^{-tA}$ and of the operator cosine family $\cos{\sqrt{A} t}$ with a constant operator $A$ possessing exponential accuracy and based on the Sinc-quadrature approximations of the corresponding Dunford-Cauchy integral representations of solutions or the solution operators.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65J10, 65M70, 35K90, 35L90

Retrieve articles in all journals with MSC (2000): 65J10, 65M70, 35K90, 35L90


Additional Information

Ivan P. Gavrilyuk
Affiliation: Berufsakademie Thüringen, Am Wartenberg 2, D-99817 Eisenach, Germany
Email: ipg@ba-eisenach.de

Volodymyr L. Makarov
Affiliation: National Academy of Sciences of Ukraine, Institute of Mathematics, Tereschen- kivska 3, 01601 Kiev, Ukraine
Email: makarov@imath.kiev.ua

DOI: https://doi.org/10.1090/S0025-5718-04-01720-X
Keywords: Evolution equation, parameter dependent operator, algorithms without accuracy saturation, exponentially convergent algorithms, Sinc-methods
Received by editor(s): January 21, 2003
Received by editor(s) in revised form: February 26, 2004
Published electronically: October 27, 2004
Article copyright: © Copyright 2004 American Mathematical Society