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Jin, X. Wen, A Hamiltonian-preserving scheme for the Liouville equation of geometric optics with partial transmissions and reflections, SIAM J. Numer.
44 (2006) 1801–1828], a Hamiltonian-preserving scheme for the Liouville equation was constructed to capture partial transmissions and reflections at the interfaces.
Previously such a problem was handled by Jin and Wen using the Liouville equation – which arises as the semiclassical limit of the Schrödinger equation – with an interface condition to account for partial transmissions and reflections (S. Jin, X. Wen, SIAM J. Num. Anal. 44 (2006) 1801–1828).
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