메뉴 건너뛰기




Volumn 50, Issue 2, 2012, Pages 492-521

Uniform error estimates of finite difference methods for the nonlinear schrödinger equation with wave operator

Author keywords

Conservative Crank Nicolson finite difference method; Energy method; Error estimates; Nonlinear Schr dinger equation with wave operator; Semi implicit finite difference method

Indexed keywords

CRANK-NICOLSON; ENERGY METHOD; ERROR ESTIMATES; SEMI-IMPLICIT; WAVE OPERATORS;

EID: 84861381433     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/110830800     Document Type: Article
Times cited : (186)

References (27)
  • 1
    • 21344479280 scopus 로고
    • Finite difference discretization of the cubic Schrödinger equation
    • G. Akrivis, Finite difference discretization of the cubic Schrödinger equation, IMA J. Numer. Anal., 13 (1993), pp. 115-124.
    • (1993) IMA J. Numer. Anal. , vol.13 , pp. 115-124
    • Akrivis, G.1
  • 2
    • 0002341433 scopus 로고
    • On fully discrete Galerkin methods of secondorder temporal accuracy for the nonlinear Schrödinger equation
    • G. Akrivis, V. Dougalis, and O. Karakashian, On fully discrete Galerkin methods of secondorder temporal accuracy for the nonlinear Schrödinger equation, Numer. Math., 59 (1991), pp. 31-53.
    • (1991) Numer. Math. , vol.59 , pp. 31-53
    • Akrivis, G.1    Dougalis, V.2    Karakashian, O.3
  • 3
    • 84861382951 scopus 로고    scopus 로고
    • Optimal error estimates of finite difference methods for the Gross-Pitaevskii equation with angular momentum rotation
    • to appear
    • W. Bao and Y. Cai, Optimal error estimates of finite difference methods for the Gross-Pitaevskii equation with angular momentum rotation, Math. Comp., to appear.
    • Math. Comp
    • Bao, W.1    Cai, Y.2
  • 4
    • 84855647334 scopus 로고    scopus 로고
    • Analysis and comparison of numerical methods for Klein-Gordon equation in nonrelativistic limit regime
    • W. Bao and X. Dong, Analysis and comparison of numerical methods for Klein-Gordon equation in nonrelativistic limit regime, Numer. Math., 120 (2012), pp. 189-229.
    • Numer. Math. , vol.120 , Issue.2012 , pp. 189-229
    • Bao, W.1    Dong, X.2
  • 5
    • 84855710301 scopus 로고    scopus 로고
    • Comparisons between sine-Gordon equation and perturbed nonlinear Schrödinger equations for modeling light bullets beyond critical collapse
    • W. Bao, X. Dong, and J. Xin, Comparisons between sine-Gordon equation and perturbed nonlinear Schrödinger equations for modeling light bullets beyond critical collapse, Phys. D, 239 (2010), pp. 1120-1134.
    • (2010) Phys. D , vol.239 , pp. 1120-1134
    • Bao, W.1    Dong, X.2    Xin, J.3
  • 6
    • 0037408314 scopus 로고    scopus 로고
    • Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation
    • DOI 10.1016/S0021-9991(03)00102-5
    • W. Bao, D. Jaksch, and P. A. Markowich, Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation, J. Comput. Phys., 187 (2003), pp. 318-342. (Pubitemid 36515583)
    • (2003) Journal of Computational Physics , vol.187 , Issue.1 , pp. 318-342
    • Bao, W.1    Jaksch, D.2    Markowich, P.A.3
  • 7
    • 0013077068 scopus 로고
    • A singular perturbation problem for an envelope equation in plasma physics
    • L. Bergé and T. Colin, A singular perturbation problem for an envelope equation in plasma physics, Phys. D, 84 (1995), pp. 437-459.
    • (1995) Phys. D , vol.84 , pp. 437-459
    • Bergé, L.1    Colin, T.2
  • 8
    • 0036556648 scopus 로고    scopus 로고
    • Order estimates in time of splitting methods for the nonlinear Schrödinger equation
    • C. Besse, B. Bidégaray, and S. Descombes, Order estimates in time of splitting methods for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., 40 (2002), pp. 26-40.
    • (2002) SIAM J. Numer. Anal. , vol.40 , pp. 26-40
    • Besse, C.1    Bidégaray, B.2    Descombes, S.3
  • 9
    • 84968490649 scopus 로고
    • Finite difference method for generalized Zakharov equations
    • Q. Chang, B. Guo, and H. Jiang, Finite difference method for generalized Zakharov equations, Math. Comp., 64 (1995), pp. 537-553.
    • (1995) Math. Comp. , vol.64 , pp. 537-553
    • Chang, Q.1    Guo, B.2    Jiang, H.3
  • 10
    • 0032384306 scopus 로고    scopus 로고
    • Semidiscretization in time for Schrödinger-waves equations
    • T. Colin and P. Fabrie, Semidiscretization in time for Schrödinger-waves equations, Discrete Contin. Dynam. Syst., 4 (1998), pp. 671-690.
    • (1998) Discrete Contin. Dynam. Syst. , vol.4 , pp. 671-690
    • Colin, T.1    Fabrie, P.2
  • 11
    • 79952290629 scopus 로고    scopus 로고
    • Modified energy for split-step methods applied to the linear Schrödinger equations
    • A. Debussche and E. Faou, Modified energy for split-step methods applied to the linear Schrödinger equations, SIAM J. Numer. Anal., 47 (2009), pp. 3705-3719.
    • (2009) SIAM J. Numer. Anal. , vol.47 , pp. 3705-3719
    • Debussche, A.1    Faou, E.2
  • 12
    • 55349136613 scopus 로고    scopus 로고
    • Analysis of an asymptotic preserving scheme for the Euler-Poisson system in the quasineutral limit
    • P. Degond, J. Liu, and M. Vignal, Analysis of an asymptotic preserving scheme for the Euler-Poisson system in the quasineutral limit, SIAM J. Numer. Anal., 46 (2008), pp. 1298-1322.
    • (2008) SIAM J. Numer. Anal. , vol.46 , pp. 1298-1322
    • Degond, P.1    Liu, J.2    Vignal, M.3
  • 13
    • 84968513694 scopus 로고
    • Convergence of an energy-preserving scheme for the Zakharov equations in one space dimension
    • R. T. Glassey, Convergence of an energy-preserving scheme for the Zakharov equations in one space dimension, Math. Comp., 58 (1992), pp. 83-102.
    • (1992) Math. Comp. , vol.58 , pp. 83-102
    • Glassey, R.T.1
  • 14
    • 0041681179 scopus 로고
    • On the problem of numerical calculation for a class of systems of nonlinear Schrödinger equations with wave operator
    • B. Guo and H. Liang, On the problem of numerical calculation for a class of systems of nonlinear Schrödinger equations with wave operator, J. Numer. Methods Comput. Appl., 4 (1983), pp. 176-182.
    • (1983) J. Numer. Methods Comput. Appl. , vol.4 , pp. 176-182
    • Guo, B.1    Liang, H.2
  • 15
    • 0001668244 scopus 로고
    • Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations
    • R. H. Hardin and F. D. Tappert, Applications of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations, SIAM Rev. Chronicle, 15 (1973), p. 423.
    • (1973) SIAM Rev. Chronicle , vol.15 , pp. 423
    • Hardin, R.H.1    Tappert, F.D.2
  • 16
    • 0033295423 scopus 로고    scopus 로고
    • Efficient Asymptotic-Preserving (AP) schemes for some multiscale kinetic equations
    • S. Jin, Efficient Asymptotic-Preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comput., 21 (1999), pp. 441-454.
    • (1999) SIAM J. Sci. Comput. , vol.21 , pp. 441-454
    • Jin, S.1
  • 17
    • 57049171510 scopus 로고    scopus 로고
    • On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations
    • C. Lubich, On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations, Math. Comp., 77 (2008), pp. 2141-2153.
    • (2008) Math. Comp. , vol.77 , pp. 2141-2153
    • Lubich, C.1
  • 18
    • 0036013587 scopus 로고    scopus 로고
    • Nonrelativistic limit in the energy space for nonlinear Klein-Gordon equations
    • DOI 10.1007/s002080200008
    • S. Machihara, K. Nakanishi, and T. Ozawa, Nonrelativistic limit in the energy space for nonlinear Klein-Gordon equations, Math. Ann., 322 (2002), pp. 603-621. (Pubitemid 36185751)
    • (2002) Mathematische Annalen , vol.322 , Issue.3 , pp. 603-621
    • Machihara, S.1    Nakanishi, K.2    Ozawa, T.3
  • 19
    • 61849133372 scopus 로고    scopus 로고
    • On the convergence of splitting methods for linear evolutionary Schrödinger equations involving an unbounded potential
    • C. Neuhauser and M. Thalhammer, On the convergence of splitting methods for linear evolutionary Schrödinger equations involving an unbounded potential, BIT, 49 (2009), pp. 199-215.
    • (2009) BIT , vol.49 , pp. 199-215
    • Neuhauser, C.1    Thalhammer, M.2
  • 20
    • 0043232734 scopus 로고    scopus 로고
    • A space-Time finite element method for the nonlinear Schrödinger equation: The continuous Galerkin method
    • K. Ohannes and M. Charalambos, A space-Time finite element method for the nonlinear Schrödinger equation: The continuous Galerkin method, SIAM J. Numer. Anal., 36 (1999), pp. 1779-1807.
    • (1999) SIAM J. Numer. Anal. , vol.36 , pp. 1779-1807
    • Ohannes, K.1    Charalambos, M.2
  • 21
    • 38249006496 scopus 로고
    • On the numerical solution of the cubic Schrödinger equation in one space variable
    • M. P. Robinson, G. Fairweather, and B. M. Herbst, On the numerical solution of the cubic Schrödinger equation in one space variable, J. Comput. Phys., 104 (1993), pp. 277-284.
    • (1993) J. Comput. Phys. , vol.104 , pp. 277-284
    • Robinson, M.P.1    Fairweather, G.2    Herbst, B.M.3
  • 22
    • 0040952168 scopus 로고
    • On the nonrelativistic limits of the Klein-Gordon and Dirac equations
    • A. Y. Schoene, On the nonrelativistic limits of the Klein-Gordon and Dirac equations, J. Math. Anal. Appl., 71 (1979), pp. 36-47.
    • (1979) J. Math. Anal. Appl. , vol.71 , pp. 36-47
    • Schoene, A.Y.1
  • 23
    • 48549114390 scopus 로고
    • Analytical and numerical aspects of certain nonlinear evolution equations, II. Numerical, nonlinear Schrödinger equation
    • SIAM J. Numer. Anal., 46 (2008), pp. 2022-2038
    • T. R. Taha and M. J. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equations, II. Numerical, nonlinear Schrödinger equation, J. Comput. Phys., 55 (1984), pp. 203-230. SIAM J. Numer. Anal., 46 (2008), pp. 2022-2038.
    • (1984) J. Comput. Phys. , vol.55 , pp. 203-230
    • Taha, T.R.1    Ablowitz, M.J.2
  • 25
    • 0001643458 scopus 로고
    • Nonrelativistic approximation of nonlinear Klein-Gordon equations in two space dimensions
    • M. Tsutumi, Nonrelativistic approximation of nonlinear Klein-Gordon equations in two space dimensions, Nonlinear Anal., 8 (1984), pp. 637-643.
    • (1984) Nonlinear Anal. , vol.8 , pp. 637-643
    • Tsutumi, M.1
  • 26
    • 33751539666 scopus 로고    scopus 로고
    • Analysis of some new conservative schemes for nonlinear Schrödinger equation with wave operator
    • DOI 10.1016/j.amc.2006.06.015, PII S009630030600525X
    • T. Wang and L. Zhang, Analysis of some new conservative schemes for nonlinear Schrödinger equation with wave operator, Appl. Math. Comput., 182 (2006), pp. 1780-1794. (Pubitemid 44838877)
    • (2006) Applied Mathematics and Computation , vol.182 , Issue.2 , pp. 1780-1794
    • Wang, T.-c.1    Zhang, L.-m.2
  • 27
    • 0346688073 scopus 로고    scopus 로고
    • Modeling light bullets with the two-dimensional sine-Gordon equation
    • J. Xin, Modeling light bullets with the two-dimensional sine-Gordon equation, Phys. D, 135 (2000), pp. 345-368.
    • (2000) Phys. D , vol.135 , pp. 345-368
    • Xin, J.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.