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

Gavrilyuk, Ivan; Makarov, Volodymyr

doi:10.1090/S0025-5718-04-01720-X

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.

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