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Numerical methods in acoustics and nonlinear dispersive partial differential equations

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If you have a question about this talk, please contact Alistair Hales.

In this talk, we will present an overview of some recent results on numerical techniques for the solution of wave scattering problems based on the boundary integral method. To begin with, we will study the solution of these integral equations using collocation methods. We will demonstrate both in practical computations and in terms of rigorous theoretical results the improved convergence properties which can be achieved with the use of least-squares oversampling. These collocation methods naturally lead to a problem of highly oscillatory quadrature because the entries in the discrete linear system representing the continuous integral equation for hybrid numerical-asymptotic basis spaces are given by singular oscillatory integrals. We develop efficient numerical methods that can compute these integrals at frequency-independent cost. Finally, based on a deep connection between oscillatory phenomena and the regularity (differentiability) of solutions to partial differential equations on periodic domains, we will see how ideas from highly oscillatory quadrature can be used in time-stepping methods to accurately capture frequency interactions in nonlinear evolution equations. Based on recent advances in resonance-based integrators, this insight allows us to design innovative numerical schemes, which can efficiently approximate low-regularity solutions to nonlinear systems, even when classical tools (such as Runge—Kutta methods) fail. This is joint work with Daan Huybrechs, Arieh Iserles, Nigel Peake and Katharina Schratz.

This talk is part of the Waves Group (DAMTP) series.

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