Some new results in multiphase geometrical optics

Olof Runborg

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 34, Issue: 6, page 1203-1231
  • ISSN: 0764-583X

Abstract

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In order to accommodate solutions with multiple phases, corresponding to crossing rays, we formulate 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 an associated 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 on delta and Heaviside functions and analyze the resulting equations. They form systems of nonlinear conservation laws with source terms. In contrast to the classical eikonal equation, these equations will incorporate a "finite" superposition principle in the sense that while the maximum number of phases is not exceeded a sum of solutions is also a solution. We present numerical results for a variety of homogeneous and inhomogeneous problems.

How to cite

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Runborg, Olof. "Some new results in multiphase geometrical optics." ESAIM: Mathematical Modelling and Numerical Analysis 34.6 (2010): 1203-1231. <http://eudml.org/doc/197596>.

@article{Runborg2010,
abstract = { In order to accommodate solutions with multiple phases, corresponding to crossing rays, we formulate 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 an associated 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 on delta and Heaviside functions and analyze the resulting equations. They form systems of nonlinear conservation laws with source terms. In contrast to the classical eikonal equation, these equations will incorporate a "finite" superposition principle in the sense that while the maximum number of phases is not exceeded a sum of solutions is also a solution. We present numerical results for a variety of homogeneous and inhomogeneous problems. },
author = {Runborg, Olof},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Geometrical optics; multivalued traveltimes; eikonal equation; kinetic equations; conservation laws; moment equations; finite difference methods; nonstrictly hyperbolic system.; multivalued travel times; finite difference methods; nonstrictly hyperbolic system; geometrical optics; scalar wave equation; kinetic transport equation; phase space; multiphase solution; nonlinear conservation laws},
language = {eng},
month = {3},
number = {6},
pages = {1203-1231},
publisher = {EDP Sciences},
title = {Some new results in multiphase geometrical optics},
url = {http://eudml.org/doc/197596},
volume = {34},
year = {2010},
}

TY - JOUR
AU - Runborg, Olof
TI - Some new results in multiphase geometrical optics
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 34
IS - 6
SP - 1203
EP - 1231
AB - In order to accommodate solutions with multiple phases, corresponding to crossing rays, we formulate 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 an associated 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 on delta and Heaviside functions and analyze the resulting equations. They form systems of nonlinear conservation laws with source terms. In contrast to the classical eikonal equation, these equations will incorporate a "finite" superposition principle in the sense that while the maximum number of phases is not exceeded a sum of solutions is also a solution. We present numerical results for a variety of homogeneous and inhomogeneous problems.
LA - eng
KW - Geometrical optics; multivalued traveltimes; eikonal equation; kinetic equations; conservation laws; moment equations; finite difference methods; nonstrictly hyperbolic system.; multivalued travel times; finite difference methods; nonstrictly hyperbolic system; geometrical optics; scalar wave equation; kinetic transport equation; phase space; multiphase solution; nonlinear conservation laws
UR - http://eudml.org/doc/197596
ER -

References

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