메뉴 건너뛰기




Volumn 56, Issue 6, 2006, Pages 814-843

Globally conservative properties and error estimation of a multi-symplectic scheme for Schrödinger equations with variable coefficients

Author keywords

Conservation laws; Error estimation; Global energy transit; Multi symplectic integrators; Schr dinger equations

Indexed keywords

ENERGY CONSERVATION; ENERGY TRANSFER; NONLINEAR EQUATIONS; NUMERICAL ANALYSIS;

EID: 33645984826     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.apnum.2005.06.006     Document Type: Article
Times cited : (70)

References (45)
  • 1
    • 33646008005 scopus 로고
    • Hamiltonian integrators for the nonlinear Schrödinger equation
    • Ablowitz M.J., and Schober C.M. Hamiltonian integrators for the nonlinear Schrödinger equation. Internat. J. Modern Phys. C 5 (1994) 397-401
    • (1994) Internat. J. Modern Phys. C , vol.5 , pp. 397-401
    • Ablowitz, M.J.1    Schober, C.M.2
  • 2
    • 0034676955 scopus 로고    scopus 로고
    • Splitting methods for the time-dependent Schrödinger equation
    • Blanes S., and Moan P.C. Splitting methods for the time-dependent Schrödinger equation. Phys. Lett. A 265 (2000) 35-42
    • (2000) Phys. Lett. A , vol.265 , pp. 35-42
    • Blanes, S.1    Moan, P.C.2
  • 3
    • 0003230767 scopus 로고    scopus 로고
    • Global Solutions of Nonlinear Schrödinger Equations
    • American Mathematical Society, Providence, RI
    • Bourgain J. Global Solutions of Nonlinear Schrödinger Equations. Amer. Math. Soc. Colloq. Publ. vol. 46 (1999), American Mathematical Society, Providence, RI
    • (1999) Amer. Math. Soc. Colloq. Publ. , vol.46
    • Bourgain, J.1
  • 4
    • 0042137401 scopus 로고    scopus 로고
    • Multi-symplectic structures and wave propagation
    • Bridges T.J. Multi-symplectic structures and wave propagation. Math. Proc. Cambridge Philos. Soc. 121 (1997) 147-190
    • (1997) Math. Proc. Cambridge Philos. Soc. , vol.121 , pp. 147-190
    • Bridges, T.J.1
  • 5
    • 0037832748 scopus 로고    scopus 로고
    • Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity
    • Bridges T.J., and Reich S. Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity. Phys. Lett. A 284 4-5 (2001) 184-193
    • (2001) Phys. Lett. A , vol.284 , Issue.4-5 , pp. 184-193
    • Bridges, T.J.1    Reich, S.2
  • 6
    • 1042300924 scopus 로고    scopus 로고
    • Geometric integration and its application
    • North-Holland, Amsterdam
    • Budd C.J., and Piggott M.D. Geometric integration and its application. Handbook Numer. Anal. vol. XI (2003), North-Holland, Amsterdam 35-139
    • (2003) Handbook Numer. Anal. , vol.XI , pp. 35-139
    • Budd, C.J.1    Piggott, M.D.2
  • 7
    • 33645976504 scopus 로고    scopus 로고
    • J. Chen, M-Z. Qin, Multi-symplectic geometry and multisymplectic integrators for the nonlinear Schrödinger equation, Preprint, 2000
  • 8
    • 0035892442 scopus 로고    scopus 로고
    • New schemes for the nonlinear Schrödinger equation
    • Chen J. New schemes for the nonlinear Schrödinger equation. Appl. Math. Comput. 124 (2001) 371-379
    • (2001) Appl. Math. Comput. , vol.124 , pp. 371-379
    • Chen, J.1
  • 9
    • 49149137309 scopus 로고
    • Finite-difference solutions of a non-linear Schrödinger equation
    • Delfour M., Fortin M., and Payre G. Finite-difference solutions of a non-linear Schrödinger equation. J. Comput. Phys. 44 (1981) 277-288
    • (1981) J. Comput. Phys. , vol.44 , pp. 277-288
    • Delfour, M.1    Fortin, M.2    Payre, G.3
  • 11
    • 0002417767 scopus 로고
    • Two-dimensional time-dependent quantum-mechanical scattering event
    • Galbraith I., Ching Y.S., and Abraham E. Two-dimensional time-dependent quantum-mechanical scattering event. Amer. J. Phys. 52 1 (1984) 60-68
    • (1984) Amer. J. Phys. , vol.52 , Issue.1 , pp. 60-68
    • Galbraith, I.1    Ching, Y.S.2    Abraham, E.3
  • 12
    • 0000074154 scopus 로고    scopus 로고
    • Symplectic integrators tailored to the time-dependent Schrödinger equation
    • Gray S.K., and Manolopoulos D.E. Symplectic integrators tailored to the time-dependent Schrödinger equation. J. Chem. Phys. 104 18 (1996) 7099-7112
    • (1996) J. Chem. Phys. , vol.104 , Issue.18 , pp. 7099-7112
    • Gray, S.K.1    Manolopoulos, D.E.2
  • 13
    • 84950807175 scopus 로고
    • Computer-generated motion pictures of one-dimensional quantum-mechanical transmission and reflection phenomena
    • Goldberg A., and Schey H.M. Computer-generated motion pictures of one-dimensional quantum-mechanical transmission and reflection phenomena. Amer. J. Phys. 35 3 (1967) 177-186
    • (1967) Amer. J. Phys. , vol.35 , Issue.3 , pp. 177-186
    • Goldberg, A.1    Schey, H.M.2
  • 14
    • 0001257175 scopus 로고
    • Numerical experiment with the nonlinear Schrödinger equation
    • Herbst B.M., Morris J., and Mitchell A.R. Numerical experiment with the nonlinear Schrödinger equation. J. Comput. Phys. 60 (1985) 282-305
    • (1985) J. Comput. Phys. , vol.60 , pp. 282-305
    • Herbst, B.M.1    Morris, J.2    Mitchell, A.R.3
  • 15
    • 0028766065 scopus 로고
    • Symplectic methods for the nonlinear Schrödinger equation
    • Herbst B.M., Varadi F., and Ablowitz M.J. Symplectic methods for the nonlinear Schrödinger equation. Math. Comput. Simulation 37 (1994) 353-369
    • (1994) Math. Comput. Simulation , vol.37 , pp. 353-369
    • Herbst, B.M.1    Varadi, F.2    Ablowitz, M.J.3
  • 16
    • 3543095217 scopus 로고    scopus 로고
    • Multi-symplecticity of the centred box discretizations for a class of Hamiltonian PDE's and an application to quasi-periodically solitary wave of qpKdV equation
    • Hong J., and Liu Y. Multi-symplecticity of the centred box discretizations for a class of Hamiltonian PDE's and an application to quasi-periodically solitary wave of qpKdV equation. Math. Comput. Model. 39 (2004) 1035-1047
    • (2004) Math. Comput. Model. , vol.39 , pp. 1035-1047
    • Hong, J.1    Liu, Y.2
  • 17
    • 84867955067 scopus 로고    scopus 로고
    • A novel numerical approach to simulating nonlinear Schrödinger equation with varying coefficients
    • Hong J., and Liu Y. A novel numerical approach to simulating nonlinear Schrödinger equation with varying coefficients. Appl. Math. Lett. 16 (2003) 759-765
    • (2003) Appl. Math. Lett. , vol.16 , pp. 759-765
    • Hong, J.1    Liu, Y.2
  • 18
    • 31244436460 scopus 로고    scopus 로고
    • Multi-symplecticity of the centred box discretizations for Hamiltonian PDE's with m {greater than or slanted equal to} 2 space dimensions
    • Hong J., and Qin M.-Z. Multi-symplecticity of the centred box discretizations for Hamiltonian PDE's with m {greater than or slanted equal to} 2 space dimensions. Appl. Math. Lett. 15 (2002) 1005-1011
    • (2002) Appl. Math. Lett. , vol.15 , pp. 1005-1011
    • Hong, J.1    Qin, M.-Z.2
  • 19
    • 0007774497 scopus 로고    scopus 로고
    • Laguerre Scheme: Another member for propagating the time-dependent Schrödinger equation
    • Hu X.-G. Laguerre Scheme: Another member for propagating the time-dependent Schrödinger equation. Phys. Rev. E 59 2 (1999) 2471-2474
    • (1999) Phys. Rev. E , vol.59 , Issue.2 , pp. 2471-2474
    • Hu, X.-G.1
  • 21
    • 0035841060 scopus 로고    scopus 로고
    • Geometric integrations for the nonlinear Schrödinger equation
    • Islas A.L., Karpeev D.A., and Schober C.M. Geometric integrations for the nonlinear Schrödinger equation. J. Comput. Phys. 173 (2001) 116-148
    • (2001) J. Comput. Phys. , vol.173 , pp. 116-148
    • Islas, A.L.1    Karpeev, D.A.2    Schober, C.M.3
  • 23
    • 0002999318 scopus 로고
    • Derivation of the discrete conservation laws for a family of finite difference schemes
    • Jiménez S. Derivation of the discrete conservation laws for a family of finite difference schemes. Appl. Math. Comput. 64 (1994) 13-45
    • (1994) Appl. Math. Comput. , vol.64 , pp. 13-45
    • Jiménez, S.1
  • 24
    • 0034918288 scopus 로고    scopus 로고
    • Symplectic integrators for discrete nonlinear Schrödinger systems
    • Karpeer D.A., and Schober C.M. Symplectic integrators for discrete nonlinear Schrödinger systems. Math. Comput. Simulation 56 (2001) 145-156
    • (2001) Math. Comput. Simulation , vol.56 , pp. 145-156
    • Karpeer, D.A.1    Schober, C.M.2
  • 25
    • 0039861863 scopus 로고    scopus 로고
    • Quantum chaos
    • Hoffmann K.H., and Schreiber M. (Eds), Springer, Berlin
    • Korsch H.J., and Wiescher H. Quantum chaos. In: Hoffmann K.H., and Schreiber M. (Eds). Computational Physics (1996), Springer, Berlin 225-244
    • (1996) Computational Physics , pp. 225-244
    • Korsch, H.J.1    Wiescher, H.2
  • 27
    • 0001529171 scopus 로고
    • Difference equations and conservation laws
    • Lee T.D. Difference equations and conservation laws. J. Statist. Phys. 46 5/6 (1987) 843-860
    • (1987) J. Statist. Phys. , vol.46 , Issue.5-6 , pp. 843-860
    • Lee, T.D.1
  • 28
    • 84965060858 scopus 로고
    • Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein-Gordon equation
    • Li S., and Vu-Quoc L. Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein-Gordon equation. SIAM J. Numer. Anal. 32 6 (1995) 1839-1875
    • (1995) SIAM J. Numer. Anal. , vol.32 , Issue.6 , pp. 1839-1875
    • Li, S.1    Vu-Quoc, L.2
  • 29
    • 0032476963 scopus 로고    scopus 로고
    • Multisymplectic geometry, variational integrators, and nonlinear PDEs
    • Marsden J.E., Patrick G.P., and Shkoller S. Multisymplectic geometry, variational integrators, and nonlinear PDEs. Comm. Math. Phys. 199 (1998) 351-395
    • (1998) Comm. Math. Phys. , vol.199 , pp. 351-395
    • Marsden, J.E.1    Patrick, G.P.2    Shkoller, S.3
  • 30
    • 0000140852 scopus 로고    scopus 로고
    • Variational methods, multisymplectic geometry and continuum mechanics
    • Marsden J.E., Pekarsky S., Shkoller S., and West M. Variational methods, multisymplectic geometry and continuum mechanics. J. Geom. Phys. 38 (2001) 253-284
    • (2001) J. Geom. Phys. , vol.38 , pp. 253-284
    • Marsden, J.E.1    Pekarsky, S.2    Shkoller, S.3    West, M.4
  • 31
    • 0013039699 scopus 로고    scopus 로고
    • Multisymplectic geometry, covariant Hamiltonians and water waves
    • Marsden J.E., and Shkoller S. Multisymplectic geometry, covariant Hamiltonians and water waves. Math. Proc. Cambridge Philos. Soc. 125 (1999) 553-575
    • (1999) Math. Proc. Cambridge Philos. Soc. , vol.125 , pp. 553-575
    • Marsden, J.E.1    Shkoller, S.2
  • 32
    • 0031270094 scopus 로고    scopus 로고
    • Optimal stability polynomials for splitting methods, with application to the time-dependent Schrödinger equation
    • McLachlan R.I., and Gray S.K. Optimal stability polynomials for splitting methods, with application to the time-dependent Schrödinger equation. Appl. Numer. Math. 25 (1997) 275-286
    • (1997) Appl. Numer. Math. , vol.25 , pp. 275-286
    • McLachlan, R.I.1    Gray, S.K.2
  • 33
    • 0000088731 scopus 로고
    • An improved acceptance procedure for the hybrid Monte Carlo algorithm
    • Neal R.M. An improved acceptance procedure for the hybrid Monte Carlo algorithm. J. Comput. Phys. 111 (1994) 194-203
    • (1994) J. Comput. Phys. , vol.111 , pp. 194-203
    • Neal, R.M.1
  • 34
    • 0040958769 scopus 로고    scopus 로고
    • A high-order accuracy method for numerical solving of the time-dependent Schrödinger equation
    • Puzyin I.V., Selin A.V., and Vinitsky S.I. A high-order accuracy method for numerical solving of the time-dependent Schrödinger equation. Comput. Phys. Comm. 123 (1999) 1-6
    • (1999) Comput. Phys. Comm. , vol.123 , pp. 1-6
    • Puzyin, I.V.1    Selin, A.V.2    Vinitsky, S.I.3
  • 35
    • 0034174119 scopus 로고    scopus 로고
    • Magnus-factorized method for numerical solving the time-dependent Schrödinger equation
    • Puzyin I.V., Selin A.V., and Vinitsky S.I. Magnus-factorized method for numerical solving the time-dependent Schrödinger equation. Comput. Phys. Comm. 126 (2000) 158-161
    • (2000) Comput. Phys. Comm. , vol.126 , pp. 158-161
    • Puzyin, I.V.1    Selin, A.V.2    Vinitsky, S.I.3
  • 36
    • 0037175405 scopus 로고    scopus 로고
    • Linearly implicit methods for the nonlinear Schrödinger equation in nonhomogeneous media
    • Ramos J.I. Linearly implicit methods for the nonlinear Schrödinger equation in nonhomogeneous media. Appl. Math. Comput. 133 (2002) 1-28
    • (2002) Appl. Math. Comput. , vol.133 , pp. 1-28
    • Ramos, J.I.1
  • 37
    • 0034687898 scopus 로고    scopus 로고
    • Multi-symplectic Runge-Kutta methods for Hamiltonian wave equations
    • Reich S. Multi-symplectic Runge-Kutta methods for Hamiltonian wave equations. J. Comput. Phys. 157 (2000) 473-499
    • (2000) J. Comput. Phys. , vol.157 , pp. 473-499
    • Reich, S.1
  • 39
    • 77957214313 scopus 로고
    • Conservative and nonconservative schemes for the solution of the nonlinear Schrödinger equation
    • Sanz-Serna J.M., and Verwer J.G. Conservative and nonconservative schemes for the solution of the nonlinear Schrödinger equation. IMA J. Numer. Anal. 6 (1986) 25-42
    • (1986) IMA J. Numer. Anal. , vol.6 , pp. 25-42
    • Sanz-Serna, J.M.1    Verwer, J.G.2
  • 40
    • 84961470712 scopus 로고
    • Methods for numerical solution of the nonlinear Schrödinger equation
    • Sanz-Serna J.M. Methods for numerical solution of the nonlinear Schrödinger equation. Math. Comp. 43 (1984) 21-27
    • (1984) Math. Comp. , vol.43 , pp. 21-27
    • Sanz-Serna, J.M.1
  • 41
    • 0001028014 scopus 로고
    • A method for the integration in time of certain partial differential equations
    • Sanz-Serna J.M., and Manoranjan V.S. A method for the integration in time of certain partial differential equations. J. Comput. Phys. 52 (1983) 273-289
    • (1983) J. Comput. Phys. , vol.52 , pp. 273-289
    • Sanz-Serna, J.M.1    Manoranjan, V.S.2
  • 42
    • 0000264271 scopus 로고    scopus 로고
    • Symplectic integrators for the Ablowitz-Ladik discrete nonlinear Schrödinger equation
    • Schober C.M. Symplectic integrators for the Ablowitz-Ladik discrete nonlinear Schrödinger equation. Phys. Lett. A 259 (1999) 140-151
    • (1999) Phys. Lett. A , vol.259 , pp. 140-151
    • Schober, C.M.1
  • 43
    • 0034319701 scopus 로고    scopus 로고
    • Novel soliton solutions of the nonlinear Schrödinger equation model
    • Serkin V.N., and Hasegawa A. Novel soliton solutions of the nonlinear Schrödinger equation model. Phys. Rev. Lett. 85 21 (2000) 4502-4505
    • (2000) Phys. Rev. Lett. , vol.85 , Issue.21 , pp. 4502-4505
    • Serkin, V.N.1    Hasegawa, A.2
  • 45
    • 0010295721 scopus 로고
    • Numerical simulation of nonlinear Schrödinger systems: A new conservative scheme
    • Zhang F., Pérez-García V.M., and Vázquez L. Numerical simulation of nonlinear Schrödinger systems: A new conservative scheme. Appl. Math. Comput. 71 (1995) 165-177
    • (1995) Appl. Math. Comput. , vol.71 , pp. 165-177
    • Zhang, F.1    Pérez-García, V.M.2    Vázquez, L.3


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.