Doctoral seminars

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Thursday, December 15th 2009, 10:30-11:30 PM in Celestijnenlaan 200A, room 05.001, 3001 Leuven-Heverlee

On asymptotic-numerical solvers for oscillatory ODEs

by Alfredo Deaño (Universidad Carlos III de Madrid). Joint work (in progress) with M. Condon (Dublin) and A. Iserles (Cambridge)

In this talk we propose a new approach to compute efficiently solutions of systems of differential equations that present highly oscillatory forcing terms. The method is based on asymptotic expansions in inverse powers of the oscillatory parameter together with modulated Fourier series for each of the coefficients in the expansion. We will discuss stability and implementation aspects of the method, and give some examples and possible directions of future research.

Numerical steepest descent with path approximations

by Andreas Asheim (Department of Mathematical Sciences, Norwegian University of Science and Technology, N-7491 Trondheim, Norway)

Combining the classical asymptotic method of steepest descent with Gaussian quadrature yields excellent quadrature methods for oscillatory integrals. In particular, they give high asymptotic accuracy, higher than comparable truncated classical asymptotic expansions. One difficulty with
these methods, when applied to certain integrals, is the computation of the paths of steepest descent. These paths are defined through non-linear
equations.
In this talk we shall see that a simple truncated power series expansions of the paths, which would not be allowed in classical asymptotic theory, can be used as approximations to the true paths. The asymptotic analysis of the resulting method is rather non-trivial and give some quite unexpected results.