Exponentially convergent algorithms for the operator exponential with applications to inhomogeneous problems in Banach spaces

by    I. P. Gavriljuk, V. Makarov

Preprint series: 04-05, Reports on Numerical Mathematics

65J10 Equations with linear operators (do not use 65Fxx)
65M70 Spectral, collocation and related methods
35K90 Abstract parabolic evolution equations
35L90 Abstract hyperbolic evolution equations

Abstract: We propose a new exponentially convergent algorithm for the operator exponential generated by a strongly positive operator A in a Banach space X. This algorithm is based on a representation by a Dunford-Cauchy integral along a path enveloping the spectrum of A combined with a proper quadrature involving a short sum of resolvents where the choose of the integration path effects dramatically desired features of the algorithms. We analyze a parabola and a hyperbola as the integration pathes and get scales of estimates in dependence on the smoothness of initial data, i. e. of the initial vector and of the inhomogeneous right-hand site. Our algorithm possesses an exponential convergence rate for the operator exponential e^-At for all t>=0 including the initial point. This allows us to construct an exponentially convergent algorithm for inhomogeneous initial value problems. It turns out that the resolvent must be modified in order to get numerically stable algorithms near the initial point. The efficiency of the proposed method is demonstrated by numerical examples.

Keywords: inhomogeneous evolution equation, operator exponential, exponentially convergent algorithms, Sinc-methods

Upload: 2004-03-16

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