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Abstract: Computation of high frequency solutions to wave equations is important inmany applications, and notoriously difficult in resolving wave oscillations.Gaussian beams are asymptotically valid high frequency solutions concentratedon a single curve through the physical domain, and superposition of Gaussianbeams provides a powerful tool to generate more general high frequencysolutions to PDEs. An alternative way to compute Gaussian beam components suchas phase, amplitude and Hessian of the phase, is to capture them in phase spaceby solving Liouville type equations on uniform grids. Following \cite{LR:2009}we present a systematic construction of asymptotic high frequency wave fieldsfrom computations in phase space for acoustic wave equations; the superpositionof phase space based Gaussian beams over two moving domains is shown necessary.Moreover, we prove that the $k$-th order Gaussian beam superposition convergesto the original wave field in the energy norm, at the rate of$\ep^{\frac{k}{2}+\frac{1-n}{4}}$ in dimension $n$.



Autor: Hailiang Liu, James Ralston

Fuente: https://arxiv.org/







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