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

Onderzoeksoutput: Bijdrage aan tijdschriftTijdschriftartikelAcademicpeer review

3 Citaties (Scopus)
8 Downloads (Pure)

Uittreksel

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.
Originele taal-2Engels
Pagina's (van-tot)739-771
Aantal pagina's33
TijdschriftJournal of Scientific Computing
Volume68
Nummer van het tijdschrift2
Vroegere onlinedatum11 feb 2016
DOI's
StatusGepubliceerd - aug 2016

Vingerafdruk

Liouville equation
Hamiltonians
Geometrical optics
Geometrical Optics
Liouville Equation
Hamiltonian Systems
Jump Conditions
Concentrator
Ray Tracing
Ray tracing
Refractive Index
Well-defined
Discontinuity
Refractive index
Jump
Absorption
Physics
Collision
Numerical Solution
First-order

Citeer dit

@article{abf62947bb1349c9b657a833a81a2bb0,
title = "A novel scheme for Liouville's equation with a discontinuous Hamiltonian and applications to geometrical optics",
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.",
author = "{van Lith}, B.S. and {ten Thije Boonkkamp}, J.H.M. and W.L. IJzerman and T.W. Tukker",
year = "2016",
month = "8",
doi = "10.1007/s10915-015-0157-6",
language = "English",
volume = "68",
pages = "739--771",
journal = "Journal of Scientific Computing",
issn = "0885-7474",
publisher = "Springer",
number = "2",

}

A novel scheme for Liouville's equation with a discontinuous Hamiltonian and applications to geometrical optics. / van Lith, B.S.; ten Thije Boonkkamp, J.H.M.; IJzerman, W.L.; Tukker, T.W.

In: Journal of Scientific Computing, Vol. 68, Nr. 2, 08.2016, blz. 739-771.

Onderzoeksoutput: Bijdrage aan tijdschriftTijdschriftartikelAcademicpeer review

TY - JOUR

T1 - A novel scheme for Liouville's equation with a discontinuous Hamiltonian and applications to geometrical optics

AU - van Lith, B.S.

AU - ten Thije Boonkkamp, J.H.M.

AU - IJzerman, W.L.

AU - Tukker, T.W.

PY - 2016/8

Y1 - 2016/8

N2 - 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.

AB - 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.

U2 - 10.1007/s10915-015-0157-6

DO - 10.1007/s10915-015-0157-6

M3 - Article

VL - 68

SP - 739

EP - 771

JO - Journal of Scientific Computing

JF - Journal of Scientific Computing

SN - 0885-7474

IS - 2

ER -