An ADER discontinuous Galerkin method on moving meshes for Liouville's equation of geometrical optics

Robert A.M. van Gestel (Corresponding author), Martijn J.H. Anthonissen, Jan H.M. ten Thije Boonkkamp, Wilbert L. IJzerman

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Liouville's equation describes light propagation through an optical system. It governs the evolution of an energy distribution on phase space. This energy distribution is discontinuous across optical interfaces. Curved optical interfaces manifest themselves as moving boundaries on phase space. In this paper, an ADER discontinuous Galerkin (DG) method on a moving mesh is applied to solve Liouville's equation. In the ADER approach a temporal Taylor series is computed by replacing temporal derivatives with spatial derivatives using the Cauchy-Kovalewski procedure. The result is a fully discrete explicit scheme of arbitrary high order of accuracy. A moving mesh is not sufficient to be able to solve Liouville's equation numerically for the optical systems considered in this article. To that end, we combine the scheme with a new method we refer to as sub-cell interface method. When dealing with optical interfaces in phase space, non-local boundary conditions arise. These are incorporated in the DG method in an energy-preserving manner. Numerical experiments validate energy-preservation up to machine precision and show the high order of accuracy. Furthermore, the DG method is compared to quasi-Monte Carlo ray tracing for two examples showing that the DG method yields better accuracy in the same amount of computational time.

Original languageEnglish
Article number112208
Number of pages28
JournalJournal of Computational Physics
Publication statusPublished - 1 Sept 2023

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).


  • ADER
  • Arbitrary-Lagrangian Eulerian
  • Discontinuous Galerkin
  • Geometrical optics
  • Liouville's equation


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