An Energy Conservative hp-method for Liouville’s Equation of Geometrical Optics

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Abstract

Liouville’s equation on phase space in geometrical optics describes the evolution of an energy distribution through an optical system, which is discontinuous across optical interfaces. The discontinuous Galerkin spectral element method is conservative and can achieve higher order of convergence locally, making it a suitable method for this equation. When dealing with optical interfaces in phase space, non-local boundary conditions arise. Besides being a difficulty in itself, these non-local boundary conditions must also satisfy energy conservation constraints. To this end, we introduce an energy conservative treatment of optical interfaces. Numerical experiments are performed to prove that the method obeys energy conservation. Furthermore, the method is compared to the industry standard ray tracing. The numerical experiments show that the discontinuous Galerkin spectral element method outperforms ray tracing by reducing the computation time by up to three orders of magnitude for an error of 10 - 6.

Original languageEnglish
Article number27
JournalJournal of Scientific Computing
Volume89
DOIs
Publication statusPublished - Oct 2021

Bibliographical note

Funding Information:
This work is part of the research programme NWO-TTW Perspectief with project number P15-36, which is (partly) financed by the Netherlands Organisation for Scientific Research (NWO).

Funding Information:
This work is funded as a part of the research programme NWO-TTW Perspectief with project number P15-36, which is (partly) financed by the Netherlands Organisation for Scientific Research (NWO).

Publisher Copyright:
© 2021, The Author(s).

Keywords

  • Discontinuous Galerkin
  • Energy conservation
  • Geometrical optics
  • Liouville’s equation
  • Phase space

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