×

A spatial sixth-order alternating direction implicit method for two-dimensional cubic nonlinear Schrödinger equations. (English) Zbl 1348.35238

Summary: Based on the combined compact difference scheme, an alternating direction implicit method is proposed for solving two-dimensional cubic nonlinear Schrödinger equations. The proposed method is sixth-order accurate in space and second-order accurate in time. The linear Fourier analysis method is exploited to study the stability of the proposed method. The efficiency and accuracy of the proposed method are tested numerically. The common solution pattern of the nonlinear Schrödinger equation is also illustrated using relevant examples known in the literature.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Chang, Q.; Jia, E.; Sun, W., Difference schemes for solving the generalized nonlinear Schrödinger equation, J. Comput. Phys., 148, 397-415 (1999) · Zbl 0923.65059
[2] Lévy, M., Parabolic Equation Methods for Electromagnetic Wave Propagations (2000), IEEE
[3] Antoine, X.; Bao, W.; Besse, C., Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations, Comput. Phys. Comm., 184, 2621-2633 (2013) · Zbl 1344.35130
[4] Bao, W.; Cai, Y., Mathematical theory and numerical methods for Bose-Einstein condensation, Kinet. Relat. Models, 6, 1-135 (2013) · Zbl 1266.82009
[5] Bao, W.; Tang, Q., Numerical study of quantized vortex interactions in the nonlinear Schrödinger equation on bounded domains, Multiscale Model. Simul., 12, 411-439 (2014) · Zbl 1320.35317
[6] Erdős, L.; Schlein, B.; Yau, H., Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems, Invent. Math., 167, 515-614 (2007) · Zbl 1123.35066
[7] Gross, E., Structure of a quantized vortex in boson systems, Neural Comput., 20, 454 (1961) · Zbl 0100.42403
[8] Kirkpatrick, K.; Schlein, B.; Staffilani, G., Derivation of the two-dimensional nonlinear Schrödinger equation from many body quantum dynamics, Amer. J. Math., 133, 91-130 (2011) · Zbl 1208.81080
[9] Zhang, Y.; Bao, W.; Du, Q., The dynamics and interaction of quantized vortices in Ginzburg-Landau-Schrödinger equations, SIAM J. Appl. Math., 67, 1740-1775 (2007) · Zbl 1220.35165
[10] Zhang, Y.; Bao, W.; Du, Q., Numerical simulation of vortex dynamics in Ginzburg-Landau-Schrödinger equation, European J. Appl. Math., 18, 607-630 (2007) · Zbl 1202.35311
[11] Dehghan, M.; Taleei, A., A compact split-step finite difference method for solving the nonlinear Schrödinger equations with constant and variable coefficients, Comput. Phys. Comm., 181, 43-51 (2010) · Zbl 1206.65207
[12] Li, X.; Zhang, L.; Wang, S., A compact finite difference scheme for the nonlinear Schrödinger equation with wave operator, Appl. Math. Comput., 219, 3187-3197 (2012) · Zbl 1309.65099
[13] Wang, S.; Zhang, L., Split-step orthogonal spline collocation methods for nonlinear Schrödinger equations in one, two, and three dimensions, Appl. Math. Comput., 218, 1903-1916 (2011) · Zbl 1231.65183
[14] Peaceman, D.; Rachford, H., The numerical solution of parabolic and elliptic differential equations, J. Soc. Ind. Appl. Math., 3, 28-41 (1955) · Zbl 0067.35801
[15] Karaa, S.; Zhang, J., High order ADI method for solving unsteady convection-diffusion problems, J. Comput. Phys., 198, 1-9 (2004) · Zbl 1053.65067
[16] Tian, Z.; Dai, S., High-order compact exponential finite difference methods for convection-diffusion type problems, Comput. Phys. Comm., 220, 952-974 (2007) · Zbl 1109.65089
[17] Tian, Z.; Ge, Y., A fourth-order compact ADI method for solving two-dimensional unsteady convection-diffusion problems, J. Comput. Appl. Math., 198, 268-286 (2007) · Zbl 1104.65086
[18] Gao, Z.; Xie, S., Fourth-order alternating direction implicit compact finite difference schemes for two-dimensional Schrödinger equation, Appl. Numer. Math., 61, 593-614 (2011) · Zbl 1221.65220
[19] Xu, Y.; Zhang, L., Alternating direction implicit method for solving two-dimensional cubic nonlinear Schrödinger equation, Comput. Phys. Comm., 183, 1082-1093 (2012) · Zbl 1277.65073
[20] Chu, P.; Fan, C., A three-point combined compact difference scheme, J. Comput. Phys., 140, 370-399 (1998) · Zbl 0923.65071
[21] Sun, H.; Li, L., A CCD-ADI method for unsteady convection-diffusion equations, Comput. Phys. Comm., 185, 790-797 (2014) · Zbl 1360.35193
[22] Fairweather, G.; Mitchell, A., A new computational procedure for ADI methods, SIAM J. Numer. Anal., 4, 163-170 (1967) · Zbl 0252.65072
[23] Chang, Q.; Xu, L., A numerical method for a system of generalized nonlinear Schrödinger equations, J. Comput. Math., 4, 191-199 (1986) · Zbl 0599.65085
[24] Bratsos, A. G., A linearized finite difference scheme for the numerical solution of the nonlinear cubic Schrödinger equation, Korean J. Comput. Appl. Math., 8, 3, 459-467 (2001) · Zbl 1015.65042
[25] Sun, H.; Zhang, J., A high-order finite difference discretization strategy based on extrapolation for convection diffusion equations, Numer. Methods Partial Differential Equations, 20, 18-32 (2004) · Zbl 1038.65108
[26] Tian, Z.; Yu, P., High-order compact ADI (HOC-ADI) method for solving unsteady 2D Schrödinger equation, Comput. Phys. Comm., 181, 861-868 (2010) · Zbl 1205.65240
[27] Liu, X.; Ding, P., Dynamic properties of cubic nonlinear Schrödinger equation with varying nonlinear parameter, J. Phys. A: Math. Gen., 37, 1589-1602 (2004) · Zbl 1046.81023
[28] Wang, T.; Guo, B.; Xu, Q., Fourth-order compact and energy conservative difference schemes for the nonlinear Schrödinger equation in two dimensions, J. Comput. Phys., 243, 382-399 (2013) · Zbl 1349.65347
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.