Algorithms without accuracy saturation for evolution equations in Hilbert and Banach spaces
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- by Ivan P. Gavrilyuk and Volodymyr L. Makarov PDF
- Math. Comp. 74 (2005), 555-583 Request permission
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
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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
- Received by editor(s): January 21, 2003
- Received by editor(s) in revised form: February 26, 2004
- Published electronically: October 27, 2004
- © Copyright 2004 American Mathematical Society
- 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
- MathSciNet review: 2114638