Adaptive methods for the Helmholtz equation with discontinuous coefficients at an interface.

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dc.contributor.advisor Sheng, Qin. Rogers, James W., Jr.
dc.contributor.other Baylor University. Dept. of Mathematics. en 2007
dc.description.abstract In this dissertation, we explore highly efficient and accurate finite difference methods for the numerical solution of variable coefficient partial differential equations arising in electromagnetic wave applications. We are particularly interested in the Helmholtz equation due to its importance in laser beam propagation simulations. The single lens environments we consider involve physical domains with subregions of differing indices of refraction. Coefficient values possess jump discontinuities at the interface between subregions. We construct novel numerical solution methods that avoid computational instability and maintain high accuracy near the interface. The first class of difference methods developed transforms the differential equation problem to a new boundary value problem for which a numerical solution can be readily computed on rectangular subregions with constant wavenumbers. The second class of numerical methods implemented combines adaptive domain transformation with coefficient smoothing to yield a boundary value problem well-suited for numerical solution on a uniform grid in the computational space. The resulting finite difference schemes do not have treat the grid points near the interface as a special case. A novel matrix analysis technique is implemented to examine the stability of these new methods. Computational verifications are carried out. en
dc.rights Baylor University theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. Contact for inquiries about permission. en
dc.subject Electromagnetic waves. en
dc.subject Helmholtz equation. en
dc.subject Differential equations. en
dc.title Adaptive methods for the Helmholtz equation with discontinuous coefficients at an interface. en
dc.type Thesis en Ph.D. en
dc.rights.accessrights Worldwide access. en
dc.rights.accessrights Access changed 5/24/11.
dc.contributor.department Mathematics. en

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