A Hamiltonian-preserving scheme for the Liouville equation of geometrical optics with partial transmissions and reflections

SIAM Journal on Numerical Analysis

Volumn 44, Issue 5, 2006, Pages 1801-1828

  Jin, Shi ^a,b   Wen, Xin ^c  

^a UNIVERSITY OF WISCONSIN MADISON   (United States)

^b TSINGHUA UNIVERSITY   (China)

^c INSTITUTE OF GEOLOGY AND GEOPHYSICS   (China)

Author keywords

Geometrical optics; Hamiltonian preserving schemes; Liouville equation; Transmission and reflection

Indexed keywords

LINEAR WAVE EQUATIONS; LIOUVILLE EQUATIONS; PARTIAL TRANSMISSIONS;

GEOMETRICAL OPTICS; LIGHT TRANSMISSION; LINEAR EQUATIONS; REFLECTION; REFRACTION; WAVE EQUATIONS;

HAMILTONIANS;

EID: 34548672178     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/050631343     Document Type: Article

References (42)

1. G. BAL, J. B. KELLER, G. PAPANICOLAOU, AND L. RYZHIK, Transport theory for acoustic waves with reflection and transmission at interfaces, Wave Motion, 30 (1999), pp. 303-327.
2. N. BEN ABDALLAH, P. DEGOND, AND I. M. GAMBA, Coupling one-dimensional time-dependent classical and quantum transport models, J. Math. Phys., 43 (2002), pp. 1-24.
3. J.-D. BENAMOU, Big ray tracing: Multivalued travel time field computation using viscosity solutions of the Eikonal equation, J. Comput. Phys., 128 (1996), pp. 463-474.
4. J.-D. BENAMOU, Direct computation of multivalued phase space solutions for Hamilton-Jacobi equations, Comm. Pure Appl. Math., 52 (1999), pp. 1443-1475.
5. J.-D. BENAMOU, An introduction to Eulerian geometrical optics (1992-2002), J. Sci. Comput., 19 (2003), pp. 63-93.
6. J.-D. BENAMOU AND I. SOLLIEC, An Eulerian method for capturing caustics, J. Comput. Phys., 162 (2000), pp. 132-163.
7. F. BOUCHUT, Renormalized solutions to the Vlasov equation with coefficients of bounded variations, Arch. Ration. Mech. Anal., 157 (2001), pp. 75-90.
8. F. BOUCHUT AND L. DESVILLETTES, On two-dimensional Hamiltonian transport equations with continuous coefficients, Differential Integral Equations, 14 (2001), pp. 1015-1024.
9. L.-T. CHENG, M. KANG, S. OSHER, H. SHIM, AND Y.-H. TSAI, Reflection in a level set framework for geometric optics, CMES Comput. Model. Eng. Sci., 5 (2004), pp. 347-360.
10. L.-T. CHENG, H. LIU, AND S. OSHER, Computational high-frequency wave propagation using the level set method, with applications to the semi-classical limit of Schrödinger equations, Commun. Math. Sci., 1 (2003), pp. 593-621.
11. R. J. DIPERNA AND P.-L. LIONS, Ordinary differential equations, transport theory, and Sobolev spaces, Invent. Math., 98 (1989), pp. 511-547.
12. B. ENGQUIST AND O. RUNBORG, Multi-phase computations in geometrical optics, J. Comput. Appl. Math., 74 (1996), pp. 175-192.
13. B. ENGQUIST AND O. RUNBORG, Multiphase computations in geometrical optics, in Hyperbolic Problems: Theory, Numerics, Applications, Internat. Ser. Numer. Math. 129, M. Fey and R. Jeltsch, eds., ETH Zentrum, Zürich, Switzerland, 1998, Birkhauser-Verlag, Basel, 1999.
14. B. ENGQUIST AND O. RUNBORG, Computational high frequency wave propagation, Acta Numer., 12 (2003), pp. 181-266.
15. B. ENGQUIST, O. RUNBORG, AND A.-K. TORNBERG, High-frequency wave propagation by the segment projection method, J. Comput. Phys., 178 (2002), pp. 373-390.
16. B. ENGQUIST, A.-K. TORNBERG, AND R. TSAI, Discretization of Dirac delta functions in level set methods, J. Comput. Phys., 207 (2005), pp. 28-51.
17. E. FATEMI, B. ENGQUIST, AND S. OSHER, Numerical solution of the high frequency asymptotic expansion for the scalar wave equation, J. Comput. Phys., 120 (1995), pp. 145-155.
18. S. FOMEL AND J. A. SETHIAN, Fast-phase space computation of multiple arrivals, Proc. Natl. Acad. Sci., 99 (2002), pp. 7329-7334.
19. L. GOSSE, Using K-branch entropy solutions for multivalued geometric optics computations, J. Comput. Phys., 180 (2002), pp. 155-182.
20. L. GOSSE, Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice. II. Impurities, confinement, and Bloch oscillations, J. Comput. Phys., 201 (2004), pp. 344-375.
21. L. GOSSE, S. JIN, AND X. T. LI, On two moment systems for computing multiphase semiclassical limits of the Schrödinger equation, Math. Model Methods Appl. Sci., 13 (2003), pp. 1689-1723.
22. loc, Comm. Partial Differential Equations, 29 (2004), pp. 207-217.
23. p coefficients, Ann. Inst. H. Poincare Anal. Non Linéaire, 20 (2003), pp. 625-644.
24. S. JIN, H. LIU, S. OSHER, AND Y.-S.R. TSAI, Computing multivalued physical observables for the semiclassical limit of the Schrödinger equation, J. Comput. Phys., 205 (2005), pp. 222-241.
25. S. JIN, H. LIU, S. OSHER, AND Y.-S.R. TSAI, Computing multi-valued physical observables for high frequency limit of symmetric hyperbolic systems, J. Comput. Phys., 210 (2005), pp. 497-518.
26. S. JIN AND S. OSHER, A level set method for the computation of multivalued solutions to quasilinear hyperbolic PDEs and Hamilton-Jacobi equations, Commun. Math. Sci., 1 (2003), pp. 575-591.
27. S. JIN AND X. WEN, Hamiltonian-preserving schemes for the Liouville equation with discontinuous potentials, Commun. Math. Sci., 3 (2005), pp. 285-315.
28. S. JIN AND X. WEN, Hamiltonian-preserving schemes for the Liouville equation of geometrical optics with discontinuous local wave speeds, J. Comput. Phys. 214 (2006), pp. 672-697.
29. 1- stability of a Hamiltonian-preserving scheme for the Liouville equation with discontinuous potentials, Math. Comp., submitted.
30. N. N. KUZNETSOV, On stable methods for solving nonlinear first order partial differential equations in the class of discontinuous functions, Topics in Numerical Analysis, III, Proc. Roy. Irish Acad. Conf., J. J. H. Miller, ed., Academic Press, London, 1977, pp. 183-197.
31. R. J. LEVEQUE, Numerical Methods for Conservation Laws, Birkhäuser-Verlag, Basel, 1990.
32. R. J. LEVEQUE AND C. ZHANG, The immersed interface method for acoustic wave equations with discontinuous coefficients, Wave Motion, 25 (1997), pp. 237-263.
33. L. MILLER, Refraction of high-frequency waves density by sharp interfaces and semiclassical measures at the boundary, J. Math. Pures Appl. (9), 79 (2000), pp. 227-269.
34. S. OSHER, L.-T. CHENG, M. KANG, H. SHIM, AND Y.-H. TSAI, Geometric optics in a phase-space-based level set and Eulerian framework, J. Comput. Phys., 179 (2002), pp. 622-648.
35. B. PERTHAME AND C.-W. SHU, On positivity preserving finite volume schemes for Euler equations, Numer. Math., 73 (1996), pp. 119-130.
36. B. PERTHAME AND C. SIMEONI, A kinetic scheme for the Saint-Venant system with a source term, Calcolo, 38 (2001), pp. 201-231.
37. O. RUNBORG, Some new results in multiphase geometrical optics, M2AN Math. Model. Numer. Anal., 34 (2000), pp. 1203-1231.
38. L. RYZHIK, G. PAPANICOLAOU, AND J. B. KELLER, Transport equations for elastic and other waves in random media, Wave Motion, 24 (1996), pp. 327-370.
39. L. RYZHIK, G. PAPANICOLAOU, AND J. B. KELLER, Transport equations for waves in a half space, Comm. Partial Differential Equations, 22 (1997), pp. 1869-1910.
40. C.-W. SHU AND S. OSHER, Efficient implementation of essentially nonoscillatory shock capturing scheme, J. Comput. Phys., 77 (1988), pp. 439-471.
41. W. W. SYMES AND J. QIAN, A slowness matching Eulerian method for multivalued solutions of Eikonal equations, J. Sci. Comput., 19 (2003), pp. 501-526. Special issue in honor of the sixtieth birthday of Stanley Osher.
42. 1 -error estimate for monotone difference schemes, Math. Comp., 64 (1995), pp. 581-589.