An adaptive spectral/DG method for a reduced phase-space based level set approach to geometrical optics on curved elements

Bernardo Cockburn, Chiu Yen Kao, Fernando Reitich

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We present an adaptive spectral/discontinuous Galerkin (DG) method on curved elements to simulate high-frequency wavefronts within a reduced phase-space formulation of geometrical optics. Following recent work, the approach is based on the use of level sets defined by functions satisfying the Liouville equations in reduced phase-space and, in particular, it relies on the smoothness of these functions to represent them by rapidly convergent spectral expansions in the phase variables. The resulting (hyperbolic) system of equations for the coefficients in these expansions are then amenable to a high-order accurate treatment via DG approximations. In the present work, we significantly expand on the applicability and efficiency of the approach by incorporating mechanisms that allow for its use in scattering simulations and for a reduced overall computational cost. With regards to the former we demonstrate that the incorporation of curved elements is necessary to attain any kind of accuracy in calculations that involve scattering off non-flat interfaces. With regards to efficiency, on the other hand, we also show that the level-set formulation allows for a space p-adaptive scheme that under-resolves the level-set functions away from the wavefront without incurring in a loss of accuracy in the approximation of its location. As we show, these improvements enable simulations that are beyond the capabilities of previous implementations of these numerical procedures.

Original languageEnglish (US)
Pages (from-to)636-649
Number of pages14
JournalJournal of Computational Physics
StatePublished - Feb 15 2014

Bibliographical note

Funding Information:
C.Y.K. would like to thank Ming-Huang Chen for fruitful discussions. C.Y.K. is partial supported by NSF DMS-1318364 . F.R. gratefully acknowledges support from NSF through grant No. DMS-0311763 .


  • Discontinuous Galerkin method
  • Eikonal equation
  • Geometrical optics
  • Level set method
  • Liouville equation
  • Spectral method
  • Wave equation


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