E 28

In the talk we present numerical methods for the approximation of two classes of operator functions that serve as propagators for the abstract parabolic and hyperbolic equations. The methods are obtained from the Dunford-Cauchy integral representation of the given operator function, via the parametrization of the integrand on the contour that envelopes the spectrum of the operator and the subsequent application of the appropriate quadrature rule. This technique reduces the operator function approximation to the sequence of independent resolvent evaluations, all of which can be done concurrently. When properly coded and run in parallel, the algorithmic implementation of the considered methods permits one to dramatically reduce the elapsed real time of computation compared to sequential implementation or other time-stepping methods. We demonstrate such comparisons on a example of the drift-diffusion model and discuss how to combine the mentioned time discretization with the convenient space discretization methods. The cases of both small and large P`eclet numbers are considered.

Dr. Dmytro Sytnyk

National Academy of Sciences of Ukraine