Volume 25, Issue 3
Initial Value Techniques for the Helmholtz and Maxwell Equations

J. Comp. Math., 25 (2007), pp. 368-373.

Published online: 2007-06

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• Abstract

We study the initial value problem of the Helmholtz equation with spatially variable wave number. We show that it can be stabilized by suppressing the evanescent waves. The stabilized Helmholtz equation can be solved numerically by a marching scheme combined with FFT. The resulting algorithm has complexity $n^2 \log n$ on a $n \times n$ grid. We demonstrate the efficacy of the method by numerical examples with caustics. For the Maxwell equation the same treatment is possible after reducing it to a second order system. We show how the method can be used for inverse problems arising in acoustic tomography and microwave imaging.

• Keywords

Stability of elliptic initial value problems, Parabolic wave equation, Inverse problems in acoustics and electromagnetics.

35J05, 65N21.

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@Article{JCM-25-368, author = {}, title = {Initial Value Techniques for the Helmholtz and Maxwell Equations}, journal = {Journal of Computational Mathematics}, year = {2007}, volume = {25}, number = {3}, pages = {368--373}, abstract = {

We study the initial value problem of the Helmholtz equation with spatially variable wave number. We show that it can be stabilized by suppressing the evanescent waves. The stabilized Helmholtz equation can be solved numerically by a marching scheme combined with FFT. The resulting algorithm has complexity $n^2 \log n$ on a $n \times n$ grid. We demonstrate the efficacy of the method by numerical examples with caustics. For the Maxwell equation the same treatment is possible after reducing it to a second order system. We show how the method can be used for inverse problems arising in acoustic tomography and microwave imaging.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8697.html} }
TY - JOUR T1 - Initial Value Techniques for the Helmholtz and Maxwell Equations JO - Journal of Computational Mathematics VL - 3 SP - 368 EP - 373 PY - 2007 DA - 2007/06 SN - 25 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/8697.html KW - Stability of elliptic initial value problems, Parabolic wave equation, Inverse problems in acoustics and electromagnetics. AB -

We study the initial value problem of the Helmholtz equation with spatially variable wave number. We show that it can be stabilized by suppressing the evanescent waves. The stabilized Helmholtz equation can be solved numerically by a marching scheme combined with FFT. The resulting algorithm has complexity $n^2 \log n$ on a $n \times n$ grid. We demonstrate the efficacy of the method by numerical examples with caustics. For the Maxwell equation the same treatment is possible after reducing it to a second order system. We show how the method can be used for inverse problems arising in acoustic tomography and microwave imaging.

Frank Natterer & Olga Klyubina. (1970). Initial Value Techniques for the Helmholtz and Maxwell Equations. Journal of Computational Mathematics. 25 (3). 368-373. doi:
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