Unconditional \(L_\infty\) convergence of a compact ADI scheme for coupled nonlinear Schrödinger system. (English) Zbl 1437.65092
A three-level compact ADI scheme is developed to solve the two-dimensional coupled nonlinear Schrödinger (CNLS) system for which, using \(L_{\infty}\) estimates, it is proved that the new scheme is second-order accurate in time variable and fourth-order accurate in space variable, and stable. Numerical experiments demonstrate that the method is highly accurate and efficient. It can be directly extended to the three-dimensional CNLS system and multi-dimensional nonlinear Schödinger-type equations.
Reviewer: Bülent Karasözen (Ankara)
MSC:
| 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 |
| 65P10 | Numerical methods for Hamiltonian systems including symplectic integrators |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
References:
| [1] | Aydin, A.; Karasözen, B., Symplectic and multisymplectic methods for the coupled nonlinear Schrödinger equations with periodic solutions, Comput. Phys. Commun., 177, 7, 566-583 (2007) · Zbl 1196.65139 |
| [2] | Aydin, A.; Karasözen, B., Multi-symplectic integration of coupled nonlinear Schrödinger system with soliton solutions, Int. J. Comput. Math., 86, 864-882 (2009) · Zbl 1165.65084 |
| [3] | Borhanifar, A.; Abazari, R., Numerical study of nonlinear Schrödinger and coupled Schrödinger equations by differential transformation method, Opt. Commun., 283, 2026-2031 (2010) |
| [4] | Chen, J.; Zhang, L., Numerical approximation of solution for the coupled nonlinear Schrödinger equations, Acta Math. Appl. Sin. Engl. Ser., 33, 2, 435-450 (2017) · Zbl 1368.65132 |
| [5] | Chien, C.; Huang, H.; Jeng, B.; Li, Z., Two-grid discretization schemes for nonlinear Schrödinger equations, J. Comput. Appl. Math., 214, 549-571 (2008) · Zbl 1135.65361 |
| [6] | Ferreira, L.; Villamizar-Roa, E., Self-similarity and asymptotic stability for coupled nonlinear Schrödinger equations in high dimensions, Physica D, 241, 534-542 (2012) · Zbl 1236.35164 |
| [7] | Gao, Y.; Mei, L., Implicit-explicit multistep methods for general two-dimensional nonlinear Schrödinger equations, Appl. Numer. Math., 109, 41-60 (2016) · Zbl 1348.65142 |
| [8] | Gauckler, L.; Lubich, C., Nonlinear Schrödinger equations and their spectral discretizations over long times, Found. Comput. Math., 10, 141-169 (2010) · Zbl 1194.37121 |
| [9] | Griffiths, D., Introduction to Quantum Mechanics (2005), Pearson Prentice Hall |
| [10] | Hasegawa, A., Optical Solitons in Fibers (1989), Springer-Verlag: Springer-Verlag Berlin |
| [11] | Hirsh, R., High order accurate difference solutions of fluid mechanics problems by a compact differencing technique, J. Comput. Phys., 19, 90-109 (1975) · Zbl 0326.76024 |
| [12] | Ismail, M., Numerical solution of coupled nonlinear Schrödinger equation by Galerkin method, Math. Comput. Simul., 78, 532-547 (2008) · Zbl 1145.65075 |
| [13] | Ismail, M., A fourth-order explicit schemes for the coupled nonlinear Schrödinger equation, Appl. Math. Comput., 196, 273-284 (2008) · Zbl 1133.65063 |
| [14] | Ismail, M.; Alamri, S., Highly accurate finite difference method for coupled nonlinear Schrödinger equation, Int. J. Comput. Math., 81, 333-351 (2004) · Zbl 1058.65090 |
| [15] | Ismail, M.; Ashi, H.; Al-Rakhemy, F., ADI method for solving the two-dimensional coupled nonlinear Schrödinger equation, (AIP Conference Proceedings, vol. 1648 (2015)), Article 050008 pp. |
| [16] | Ismail, M.; Taha, T., A linearly implicit conservative scheme for the coupled nonlinear Schrödinger equation, Math. Comput. Simul., 74, 302-311 (2007) · Zbl 1112.65079 |
| [17] | Li, J.; Sun, Z.; Zhao, X., A three level linearized compact difference scheme for the Cahn-Hilliard equation, Sci. China Math., 55, 805-826 (2012) · Zbl 1262.65106 |
| [18] | Liao, H.; Sun, Z., A two-level compact ADI method for solving second-order wave equations, Int. J. Comput. Math., 90, 7, 1471-1488 (2013) · Zbl 1279.65099 |
| [19] | Liao, H.; Sun, Z.; Shi, H.; Wang, T., Convergence of compact ADI method for solving linear Schrödinger equations, Numer. Methods Partial Differ. Equ., 28, 5, 1598-1619 (2012) · Zbl 1259.65135 |
| [20] | Mei, C., The Applied Dynamics of Ocean Waves (1989), World Scientific: World Scientific Singapore · Zbl 0991.76003 |
| [21] | Ohta, M., Stability of solitary waves for coupled nonlinear Schrödinger equations, Nonlinear Anal., 26, 933-939 (1996) · Zbl 0860.35011 |
| [22] | Sun, J.; Qin, M., Multi-symplectic methods for the coupled 1D nonlinear Schrödinger system, Comput. Phys. Commun., 155, 221-235 (2003) · Zbl 1196.65195 |
| [23] | Sun, W.; Wang, J., Optimal error analysis of Crank-Nicolson schemes for a coupled nonlinear Schrödinger system in 3D, J. Comput. Appl. Math., 317, 685-699 (2017) · Zbl 1357.65148 |
| [24] | Sun, Z., Numerical Methods of the Partial Differential Equations (2005), Science Press: Science Press Beijing |
| [25] | Sun, Z.; Zhao, D., On the \(L_\infty\) convergence of a difference scheme for coupled nonlinear Schrödinger equations, Comput. Math. Appl., 59, 3286-3300 (2010) · Zbl 1198.65173 |
| [26] | Tian, Z.; Yu, P., High-order compact ADI (HOC-ADI) method for solving unsteady 2D Schrödinger equation, Comput. Phys. Commun., 181, 861-868 (2010) · Zbl 1205.65240 |
| [27] | Wang, T.; Guo, B.; Zhang, L., New conservative difference schemes for a coupled nonlinear Schrödinger system, Appl. Math. Comput., 217, 1604-1619 (2010) · Zbl 1205.65242 |
| [28] | Wang, T.; Nie, T.; Zhang, L.; Chen, F., Numerical simulation of nonlinearly coupled Schrödinger systems: a linearly uncoupled finite difference scheme, Math. Comput. Simul., 79, 607-621 (2008) · Zbl 1202.65116 |
| [29] | Wang, T.; Zhao, X.; Jiang, J., Unconditional and optimal \(H^2\)-error estimates of two linear and conservative finite difference schemes for the Klein-Gordon-Schrödinger equation in high dimensions, Adv. Comput. Math., 44, 477-503 (2018) · Zbl 1448.65122 |
| [30] | Yew, A., Stability analysis of multipulses in nonlinear Schrödinger equations, Indiana Univ. Math. J., 49, 1079-1124 (2000) · Zbl 0978.35057 |
| [31] | Zhou, Y., Application of Discrete Functional Analysis to the Finite Difference Method (1990), International Academic Publishers |
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