Asymptotic solutions for high frequency Helmholtz equations

dc.contributor.advisor Songting Luo Jacobs, Matthew
dc.contributor.department Mathematics 2020-09-23T19:12:47.000 2021-02-25T21:34:34Z 2021-02-25T21:34:34Z Sat Aug 01 00:00:00 UTC 2020 2020-09-10 2020-01-01
dc.description.abstract <p>In this thesis, we will investigate and develop asymptotic methods for numerically solving high frequency Helmholtz equations with point-source conditions. Due to the oscillatory nature of the wave, such equations are highly challenging to solve by conventional methods, such as the finite difference and finite element methods, since they often suffer from a large number of degrees of freedom to avoid the `pollution effect' (large dispersion errors). We shall first apply the geometrical optics (GO) approximation to compute the wave locally near the primary source, where instead of computing the oscillatory wave directly, its phase and amplitudes are computed through the eikonal equation and a recurrent system of transport equations, respectively, and are used to reconstruct the wave for any high frequencies. The GO approximation is efficient for providing locally valid approximations of the wave. We propose to further propagate the wave to the whole domain of interest through an appropriate time-dependent Schr\"{o}dinger equation whose steady-state solution in the domain of interest will provide globally valid approximations of the wave. The wavefunction of the Schr\"{o}dinger equation can be propagated by a Strang operator splitting based pseudo-spectral method that is unconditionally stable, which allows large time step sizes to reach the steady state efficiently. In the pseudo-spectral method, wherever the matrix exponential is involved, the Krylov subspace method can be used to compute the relevant matrix-vector products. The proposed asymptotic method will be effective since: (1) it is able to obtain globally valid approximations of the wave, (2) it has complexity $O(N\log N)$ where $N$ is the total number of simulation points for a prescribed accuracy requirement, and (3) the number of simulation points per wavelength can be fixed as the frequency increases. Numerical experiments in both two- and three-dimensional spaces will be performed to demonstrate the method.</p>
dc.format.mimetype application/pdf
dc.identifier archive/
dc.identifier.articleid 9158
dc.identifier.contextkey 19236704
dc.identifier.s3bucket isulib-bepress-aws-west
dc.identifier.submissionpath etd/18151
dc.language.iso en
dc.source.bitstream archive/|||Fri Jan 14 21:37:39 UTC 2022
dc.subject.keywords Anisotropic Helmholtz equation
dc.subject.keywords Babich's expansion
dc.subject.keywords Geometrical optics
dc.subject.keywords Pseudo-spectral method
dc.subject.keywords Strang operator splitting
dc.subject.keywords Time-dependent Schr\"{o}dinger equation
dc.title Asymptotic solutions for high frequency Helmholtz equations
dc.type article
dc.type.genre dissertation
dspace.entity.type Publication
relation.isOrgUnitOfPublication 82295b2b-0f85-4929-9659-075c93e82c48 Applied Mathematics dissertation Doctor of Philosophy
Original bundle
Now showing 1 - 1 of 1
No Thumbnail Available
17.7 MB
Adobe Portable Document Format