A novel scheme for Liouville's equation with a discontinuous Hamiltonian and applications to geometrical optics

B.S. van Lith, J.H.M. ten Thije Boonkkamp, W.L. IJzerman, T.W. Tukker

Research output: Contribution to journalArticleAcademicpeer-review

5 Citations (Scopus)
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Abstract

A novel scheme is developed that computes numerical solutions of Liouville’s equation with a discontinuous Hamiltonian. It is assumed that the underlying Hamiltonian system has well-defined behaviour even when the Hamiltonian is discontinuous. In the case of geometrical optics such a discontinuity yields the familiar Snell’s law or the law of specular reflection. Solutions to Liouville’s equation should be constant along curves defined by the Hamiltonian system when the right-hand side is zero, i.e., no absorption or collisions. This consideration allows us to derive a new jump condition, enabling us to construct a first-order accurate scheme. Essentially, the correct physics is built into the solver. The scheme is tested in a two-dimensional optical setting with two test cases, the first using a single jump in the refractive index and the second a compound parabolic concentrator. For these two situations, the scheme outperforms the more conventional method of Monte Carlo ray tracing.
Original languageEnglish
Pages (from-to)739-771
Number of pages33
JournalJournal of Scientific Computing
Volume68
Issue number2
Early online date11 Feb 2016
DOIs
Publication statusPublished - Aug 2016

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