메뉴 건너뛰기




Volumn 214, Issue 2, 2006, Pages 672-697

Hamiltonian-preserving schemes for the Liouville equation of geometrical optics with discontinuous local wave speeds

Author keywords

Geometrical optics; Hamiltonian preserving schemes; Level set method; Liouville equation; Snell's law of refraction

Indexed keywords

GEOMETRY; HAMILTONIANS; LEVEL MEASUREMENT; NUMERICAL METHODS; PHASE SPACE METHODS; REFRACTION;

EID: 33644952544     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2005.10.012     Document Type: Article
Times cited : (37)

References (42)
  • 1
    • 0033363402 scopus 로고    scopus 로고
    • Transport theory for acoustic waves with reflection and transmission at interfaces
    • G. Bal J.B. Keller G. Papanicolaou L. Ryzhik Transport theory for acoustic waves with reflection and transmission at interfaces Wave Motion 30 1999 303-327
    • (1999) Wave Motion , vol.30 , pp. 303-327
    • Bal, G.1    Keller, J.B.2    Papanicolaou, G.3    Ryzhik, L.4
  • 2
    • 0030588020 scopus 로고    scopus 로고
    • Big ray tracing: Multivalued travel time field computation using viscosity solutions of the Eikonal equation
    • J.-D. Benamou Big ray tracing: Multivalued travel time field computation using viscosity solutions of the Eikonal equation J. Comput. Phys. 128 1996 463-474
    • (1996) J. Comput. Phys. , vol.128 , pp. 463-474
    • Benamou, J.-D.1
  • 3
    • 0033421722 scopus 로고    scopus 로고
    • Direct computation of multivalued phase space solutions for Hamilton-Jacobi equations
    • J.-D. Benamou Direct computation of multivalued phase space solutions for Hamilton-Jacobi equations Commun. Pure Appl. Math. 52 11 1999 1443-1475
    • (1999) Commun. Pure Appl. Math. , vol.52 , Issue.11 , pp. 1443-1475
    • Benamou, J.-D.1
  • 4
    • 0345829286 scopus 로고    scopus 로고
    • An introduction to Eulerian geometrical optics (1992-2002)
    • J.-D. Benamou An introduction to Eulerian geometrical optics (1992-2002) J. Sci. Comp. 19 1-3 2001 63-93
    • (2001) J. Sci. Comp. , vol.19 , Issue.1-3 , pp. 63-93
    • Benamou, J.-D.1
  • 5
    • 0001415492 scopus 로고    scopus 로고
    • An Eulerian method for capturing caustics
    • J.-D. Benamou I. Solliec An Eulerian method for capturing caustics J. Comput. Phys. 162 1 2000 132-163
    • (2000) J. Comput. Phys. , vol.162 , Issue.1 , pp. 132-163
    • Benamou, J.-D.1    Solliec, I.2
  • 6
    • 0032092187 scopus 로고    scopus 로고
    • One-dimensional transport equations with discontinuous coefficients
    • F. Bouchut F. James One-dimensional transport equations with discontinuous coefficients Nonlinear Anal. Theory Meth. Appl. 32 1998 891-933
    • (1998) Nonlinear Anal. Theory Meth. Appl. , vol.32 , pp. 891-933
    • Bouchut, F.1    James, F.2
  • 8
    • 3242697504 scopus 로고    scopus 로고
    • Computational high-frequency wave propagation using the level set method, with applications to the semi-classical limit of Schrödinger equations
    • L.-T. Cheng H.-L. Liu 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 593-621
    • (2003) Commun. Math. Sci. , vol.1 , pp. 593-621
    • Cheng, L.-T.1    Liu, H.-L.2    Osher, S.3
  • 9
    • 0000563828 scopus 로고    scopus 로고
    • On some analogy between different approaches to first order PDEs with non-smooth coefficients
    • I. Capuzzo Dolcetta B. Perthame On some analogy between different approaches to first order PDEs with non-smooth coefficients Adv. Math. Sci. Appl 1996
    • (1996) Adv. Math. Sci. Appl
    • Capuzzo Dolcetta, I.1    Perthame, B.2
  • 10
    • 0030571228 scopus 로고    scopus 로고
    • Multi-phase computations in geometrical optics
    • B. Engquist O. Runborg Multi-phase computations in geometrical optics J. Comput. Appl. Math. 74 1996 175-192
    • (1996) J. Comput. Appl. Math. , vol.74 , pp. 175-192
    • Engquist, B.1    Runborg, O.2
  • 12
    • 24644487123 scopus 로고    scopus 로고
    • Computational high frequency wave propagation
    • B. Engquist O. Runborg Computational high frequency wave propagation Acta Numer. 12 2003 181-266
    • (2003) Acta Numer. , vol.12 , pp. 181-266
    • Engquist, B.1    Runborg, O.2
  • 13
    • 0037141033 scopus 로고    scopus 로고
    • High-frequency wave propagation by the segment projection method
    • B. Engquist O. Runborg A.-K. Tornberg High-frequency wave propagation by the segment projection method J. Comput. Phys. 178 2 2002 373-390
    • (2002) J. Comput. Phys. , vol.178 , Issue.2 , pp. 373-390
    • Engquist, B.1    Runborg, O.2    Tornberg, A.-K.3
  • 14
    • 0000647149 scopus 로고
    • Numerical solution of the high frequency asymptotic expansion for the scalar wave equation
    • E. Fatemi B. Engquist S. Osher Numerical solution of the high frequency asymptotic expansion for the scalar wave equation J. Comput. Phys. 120 1995 145-155
    • (1995) J. Comput. Phys. , vol.120 , pp. 145-155
    • Fatemi, E.1    Engquist, B.2    Osher, S.3
  • 15
    • 0001715056 scopus 로고
    • The Hamiltonian way for computing Hamiltonian dynamics
    • Applied and Industrial Mathematics, Venice 1989 Kluwer Academic Publishers Dordrecht
    • K. Feng The Hamiltonian way for computing Hamiltonian dynamics Applied and Industrial Mathematics, Venice 1989 Math. Appl. vol. 56 1991 Kluwer Academic Publishers Dordrecht pp. 17-35
    • (1991) Math. Appl. , vol.56 , pp. 17-35
    • Feng, K.1
  • 16
    • 0037188495 scopus 로고    scopus 로고
    • Fast phase space computation of multiple arrivals
    • S. Fomel J.A. Sethian Fast phase space computation of multiple arrivals Proc. Natl. Acad. Sci. USA 99 11 2002 7329-7334
    • (2002) Proc. Natl. Acad. Sci. USA , vol.99 , Issue.11 , pp. 7329-7334
    • Fomel, S.1    Sethian, J.A.2
  • 17
    • 0037143007 scopus 로고    scopus 로고
    • Using K-branch entropy solutions for multivalued geometric optics computations
    • L. Gosse Using K-branch entropy solutions for multivalued geometric optics computations J. Comput. Phys. 180 2002 155-182
    • (2002) J. Comput. Phys. , vol.180 , pp. 155-182
    • Gosse, L.1
  • 18
    • 13844269242 scopus 로고    scopus 로고
    • Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice II. Impurities, confinement and Bloch oscillations
    • 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 344-375
    • (2004) J. Comput. Phys. , vol.201 , pp. 344-375
    • Gosse, L.1
  • 19
    • 0034401759 scopus 로고    scopus 로고
    • Numerical approximations of one-dimensional linear conservation equations with discontinuous coefficients
    • L. Gosse F. James Numerical approximations of one-dimensional linear conservation equations with discontinuous coefficients Math. Comp. 2000 987-1015
    • (2000) Math. Comp. , pp. 987-1015
    • Gosse, L.1    James, F.2
  • 20
    • 0442275913 scopus 로고    scopus 로고
    • On two moment systems for computing multiphase semiclassical limits of the Schrodinger equation math
    • L. Gosse S. Jin X.T. Li On two moment systems for computing multiphase semiclassical limits of the Schrodinger equation math Model Methods Appl. Sci. 13 2003 1689-1723
    • (2003) Model Methods Appl. Sci. , vol.13 , pp. 1689-1723
    • Gosse, L.1    Jin, S.2    Li, X.T.3
  • 21
    • 16844373042 scopus 로고    scopus 로고
    • Computing multivalued physical observables for the semiclassical limit of the Schrodinger equation
    • S. Jin H.L. Liu S. Osher R. Tsai Computing multivalued physical observables for the semiclassical limit of the Schrodinger equation J. Comput. Phys. 205 2005 222-241
    • (2005) J. Comput. Phys. , vol.205 , pp. 222-241
    • Jin, S.1    Liu, H.L.2    Osher, S.3    Tsai, R.4
  • 22
    • 23944519965 scopus 로고    scopus 로고
    • Computing multi-valued physical observables for high frequency limit of symmetric hyperbolic systems
    • S. Jin H.L. Liu S. Osher R. Tsai Computing multi-valued physical observables for high frequency limit of symmetric hyperbolic systems J. Comput. Phys. 210 2005 497-518
    • (2005) J. Comput. Phys. , vol.210 , pp. 497-518
    • Jin, S.1    Liu, H.L.2    Osher, S.3    Tsai, R.4
  • 23
    • 3242673880 scopus 로고    scopus 로고
    • A level set method for the computation of multi-valued solutions to quasi-linear hyperbolic PDE's and Hamilton-Jacobi equations
    • S. Jin S. Osher A level set method for the computation of multi-valued solutions to quasi-linear hyperbolic PDE's and Hamilton-Jacobi equations Commun. Math. Sci. 1 3 2003 575-591
    • (2003) Commun. Math. Sci. , vol.1 , Issue.3 , pp. 575-591
    • Jin, S.1    Osher, S.2
  • 24
    • 33644957151 scopus 로고    scopus 로고
    • Hamiltonian-preserving schemes for the Liouville equation with discontinuous potentials
    • S. Jin X. Wen Hamiltonian-preserving schemes for the Liouville equation with discontinuous potentials Commun. Math. Sci. 3 2005 285-315
    • (2005) Commun. Math. Sci. , vol.3 , pp. 285-315
    • Jin, S.1    Wen, X.2
  • 25
    • 33644932507 scopus 로고    scopus 로고
    • 1-stability of a Hamiltonian-preserving scheme for the Liouville equation with discontinuous potentials
    • preprint
    • 1-stability of a Hamiltonian-preserving scheme for the Liouville equation with discontinuous potentials, preprint.
    • Jin, S.1    Wen, X.2
  • 26
    • 85190812400 scopus 로고    scopus 로고
    • Hamiltonian-preserving schemes for the Liouville equation of geometrical optics partial with transmissions and reflections
    • submitted
    • S. Jin, X. Wen, Hamiltonian-preserving schemes for the Liouville equation of geometrical optics partial with transmissions and reflections, SIAM J. Num. Anal., submitted.
    • SIAM J. Num. Anal.
    • Jin, S.1    Wen, X.2
  • 27
    • 0040452394 scopus 로고
    • On stable methods for solving non-linear first order partial differential equations in the class of discontinuous functions
    • J.J.H. Miller Academic Press London
    • N.N. Kuznetsov On stable methods for solving non-linear first order partial differential equations in the class of discontinuous functions J.J.H. Miller Topics in Numerical Analysis III (Proc. Roy. Irish Acad. Conf.) 1977 Academic Press London 183-197
    • (1977) Topics in Numerical Analysis III (Proc. Roy. Irish Acad. Conf.) , pp. 183-197
    • Kuznetsov, N.N.1
  • 30
    • 0034148496 scopus 로고    scopus 로고
    • Refraction of high frequency waves density by sharp interfaces and semiclassical measures at the boundary
    • L. Miller Refraction of high frequency waves density by sharp interfaces and semiclassical measures at the boundary J. Math. Pures Appl. IX 79 2000 227-269
    • (2000) J. Math. Pures Appl. IX , vol.79 , pp. 227-269
    • Miller, L.1
  • 31
    • 0036647932 scopus 로고    scopus 로고
    • Geometric optics in a phase-space-based level set and Eulerian framework
    • S. Osher L.-T. Cheng M. Kang H. Shim Y.-H. Tsai Geometric optics in a phase-space-based level set and Eulerian framework J. Comput. Phys. 179 2 2002 622-648
    • (2002) J. Comput. Phys. , vol.179 , Issue.2 , pp. 622-648
    • Osher, S.1    Cheng, L.-T.2    Kang, M.3    Shim, H.4    Tsai, Y.-H.5
  • 32
    • 0030530384 scopus 로고    scopus 로고
    • On positivity preserving finite volume schemes for Euler equations
    • B. Perthame C.W. Shu On positivity preserving finite volume schemes for Euler equations Numer. Math. 73 1996 119-130
    • (1996) Numer. Math. , vol.73 , pp. 119-130
    • Perthame, B.1    Shu, C.W.2
  • 33
    • 0012588539 scopus 로고    scopus 로고
    • A kinetic scheme for the Saint-Venant system with a source term
    • B. Perthame C. Simeoni A kinetic scheme for the Saint-Venant system with a source term CALCOLO 38 4 2001 201-231
    • (2001) CALCOLO , vol.38 , Issue.4 , pp. 201-231
    • Perthame, B.1    Simeoni, C.2
  • 34
    • 0035882842 scopus 로고    scopus 로고
    • Linear transport equations with discontinuous coefficients
    • G. Petrova B. Popov Linear transport equations with discontinuous coefficients J. Math. Anal. Appl. 260 2001 307-324
    • (2001) J. Math. Anal. Appl. , vol.260 , pp. 307-324
    • Petrova, G.1    Popov, B.2
  • 35
    • 0000389658 scopus 로고    scopus 로고
    • Measure solutions to the linear multidimensional transport equation with non-smooth coefficients
    • F. Poupaud M. Rascle Measure solutions to the linear multidimensional transport equation with non-smooth coefficients Commun. PDEs 22 1997 337-358
    • (1997) Commun. PDEs , vol.22 , pp. 337-358
    • Poupaud, F.1    Rascle, M.2
  • 36
    • 0037379052 scopus 로고    scopus 로고
    • A level set based Eulerian approach for anisotropic wave propagations
    • J. Qian L.-T. Cheng S. Osher A level set based Eulerian approach for anisotropic wave propagations Wave Motion 37 2003 365-379
    • (2003) Wave Motion , vol.37 , pp. 365-379
    • Qian, J.1    Cheng, L.-T.2    Osher, S.3
  • 37
    • 0034551460 scopus 로고    scopus 로고
    • Some new results in multiphase geometrical optics
    • O. Runborg Some new results in multiphase geometrical optics M2AN Math. Model. Numer. Anal. 34 2000 1203-1231
    • (2000) M2AN Math. Model. Numer. Anal. , vol.34 , pp. 1203-1231
    • Runborg, O.1
  • 38
    • 0030388835 scopus 로고    scopus 로고
    • Transport equations for elastic and other waves in random media
    • L. Ryzhik G. Papanicolaou J. Keller Transport equations for elastic and other waves in random media Wave Motion 24 1996 327-370
    • (1996) Wave Motion , vol.24 , pp. 327-370
    • Ryzhik, L.1    Papanicolaou, G.2    Keller, J.3
  • 39
    • 0001234945 scopus 로고    scopus 로고
    • Transport equations for waves in a half space
    • L. Ryzhik G. Papanicolaou J. Keller Transport equations for waves in a half space Commun. PDE's 22 1997 1869-1910
    • (1997) Commun. PDE's , vol.22 , pp. 1869-1910
    • Ryzhik, L.1    Papanicolaou, G.2    Keller, J.3
  • 40
    • 45449125925 scopus 로고
    • Efficient implementation of essentially non-oscillatory shock capturing scheme
    • C.W. Shu S. Osher Efficient implementation of essentially non-oscillatory shock capturing scheme J. Comput. Phys. 77 1988 439-471
    • (1988) J. Comput. Phys. , vol.77 , pp. 439-471
    • Shu, C.W.1    Osher, S.2
  • 41
    • 0347900513 scopus 로고    scopus 로고
    • A slowness matching Eulerian method for multivalued solutions of Eikonal equations
    • (special issue in honor of the sixtieth birthday of Stanley Osher)
    • W. Symes J. Qian A slowness matching Eulerian method for multivalued solutions of Eikonal equations J. Sci. Comp. 19 1-3 2003 501-526 (special issue in honor of the sixtieth birthday of Stanley Osher)
    • (2003) J. Sci. Comp. , vol.19 , Issue.1-3 , pp. 501-526
    • Symes, W.1    Qian, J.2
  • 42
    • 84968519569 scopus 로고
    • 1-error estimate for monotone difference schemes
    • 1-error estimate for monotone difference schemes Math. Comp. 64 1995 581-589
    • (1995) Math. Comp. , vol.64 , pp. 581-589
    • Tang, T.1    Teng, Z.H.2


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.