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Multiphase Computations in Geometrical Optics

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Speaker: Olof C. Runborg, Program in Applied and Computational Mathematics, Princeton University
Location: 693 Kerr
Start time: Tue, Feb 6 2001, 4:10PM

Ray tracing has long been the numerical method of choice for geometrical optics problems. For several reasons, including efficiency concerns, it has recently been challenged by PDE methods based on the non-linear eikonal and transport equations. For this formulation, however, there is no superposition principle and it cannot accomodate solutions with multiple phases, corresponding to crossing rays. We reformulate geometrical optics for the scalar wave equation as a kinetic transport equation set in phase space. If the maximum number of phases is finite we can recover the exact multiphase solution from an associated system of moment equations, closed by an assumption about the form of the density function in the kinetic equation. The moment equations form a hyperbolic system of conservation laws with source terms. Unlike the eikonal equation, the equations will incorporate a finite superposition principle in the sense that if the maximum number of phases is not exceeded, a sum of solutions is also a solution. We present numerical results of how the equations perform on a variety of problems with constant and variable index of refraction.