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Some new results in multiphase geometrical optics

Published online by Cambridge University Press:  15 April 2002

Olof Runborg
Olof Runborg*
Affiliation:
Program in Applied and Computational Mathematics, Fine Hall, Princeton University, Princeton, NJ 08544, USA. ([email protected])
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Abstract

In order to accommodate solutions with multiple phases, corresponding to crossing rays, weformulate geometrical optics for the scalar wave equation as a kinetic transport equation set in phase space. If the maximum number of phases is finite and known a priori we can recover the exact multiphase solution from anassociated system of moment equations, closed by an assumption on the form of the density function in the kinetic equation. We consider two different closure assumptions based ondelta and Heaviside functions and analyze the resultingequations. They form systems of nonlinear conservation laws with source terms. In contrast to the classicaleikonal equation, theseequations will incorporate a "finite" superposition principle in the sense that while the maximum number of phasesis not exceeded a sum of solutions is also a solution. We present numerical results for a varietyof homogeneous and inhomogeneous problems.

Keywords

Geometrical opticsmultivalued traveltimeseikonal equationkinetic equationsconservation lawsmoment equationsfinite difference methodsnonstrictly hyperbolic system.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 6 , November 2000 , pp. 1203 - 1231
Copyright
© EDP Sciences, SMAI, 2000

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