Efficient and accurate numerical methods for the Klein-Gordon-Schrödinger equations. (English) Zbl 1125.65093
The authors consider the equation named in the title, along with periodic boundary conditions, and start by listing or establishing invariance, conservation and asymptotic results. Their numerical methods contain time splittings, the partial steps of which can be solved analytically (meaning computation of a lot of exponential functions) or be approximated by Fourier methods in combination with analytic or time stepping solving. The invariance, conservation and stability properties of the numerical solutions are also investigated. A series of numerical experiments (e.g., solitary wave collisions in 1D and 2D) illustrates the methods; there are no data on computing times or 3D problems.
Reviewer: Gisbert Stoyan (Budapest)
MSC:
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
time splitting; stability; Fourier methods; numerical experiments; solitary wave collisionsReferences:
| [1] | Bao, W.; Jaksch, D., An explicit unconditionally stable numerical method for solving damped nonlinear Schrödinger equations with a focusing nonlinearity, SIAM J. Numer. Anal., 41, 1406-1426 (2003) · Zbl 1054.35088 |
| [2] | Bao, W.; Jaksch, D.; Markowich, P. A., Numerical solution of the Gross-Pitaevsikii equation for Bose-Einstein condensation, J. Comput. Phys., 187, 318-342 (2003) · Zbl 1028.82501 |
| [3] | Bao, W.; Jin, S.; Markowich, P. A., On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime, J. Comput. Phys., 175, 487-524 (2002) · Zbl 1006.65112 |
| [4] | Bao, W.; Li, X. G., An efficient and stable numerical method for the Maxwell-Dirac system, J. Comput. Phys., 199, 663-687 (2004) · Zbl 1056.81080 |
| [5] | Bao, W.; Sun, F. F., Efficient and stable numerical methods for the generalized and vector Zakharov system, SIAM J. Sci. Comput., 26, 1057-1088 (2005) · Zbl 1076.35114 |
| [6] | Bao, W.; Zheng, C., A time-splitting spectral method for three-wave interactions in media with competing quadratic and cubic nonlinearities, Commun. Comput. Phys., 2, 123-140 (2007) · Zbl 1164.78320 |
| [7] | P. Bechouche, N.J. Mauser, S. Selberg, Non-relativistic limit of Klein-Gordon-Maxerll, preprint.; P. Bechouche, N.J. Mauser, S. Selberg, Non-relativistic limit of Klein-Gordon-Maxerll, preprint. · Zbl 1064.35198 |
| [8] | P. Bechouche, N.J. Mauser, S. Selberg, Non-relativistic limit of Klein-Gordon-Maxerll to Schrödinger-Poisson, preprint.; P. Bechouche, N.J. Mauser, S. Selberg, Non-relativistic limit of Klein-Gordon-Maxerll to Schrödinger-Poisson, preprint. · Zbl 1064.35198 |
| [9] | Biler, P., Attractors for the system of Schrödinger and Klein-Gordon equations with Yukawa coupling, SIAM J. Math. Anal., 21, 1190-1212 (1990) · Zbl 0725.35020 |
| [10] | D.I. Choi, Numerical Studies of Nonlinear Schrödinger and Klein-Gordon Systems: Techniques and Applications, Ph.D. Thesis, The University of Texas at Austin, 1998.; D.I. Choi, Numerical Studies of Nonlinear Schrödinger and Klein-Gordon Systems: Techniques and Applications, Ph.D. Thesis, The University of Texas at Austin, 1998. |
| [11] | Cirincione, R. J.; Chernoff, P. R., Dirac and Klein-Gordon equations: convergence of solutions in the nonrelativistic limit, Commun. Math. Phys., 79, 33-46 (1981) · Zbl 0471.35024 |
| [12] | Darwish, A.; Fan, E. G., A series of new explicit exact solutions for the coupled Klein-Gordon-Schrödinger equations, Chaos Soliton Fract, 20, 609-617 (2004) · Zbl 1049.35167 |
| [13] | Fornberg, B.; Driscoll, T. A., A fast spectral algorithm for nonlinear wave equations with linear dispersion, J. Comput. Phys., 155, 456-467 (1999) · Zbl 0937.65109 |
| [14] | Fukuda, I.; Tsutsumi, M., On the Yukawa-coupled Klein-Gordon-Schrödinger equations in three space dimensions, Prof. Jpn. Acad., 51, 402-405 (1975) · Zbl 0313.35065 |
| [15] | Fukuda, I.; Tsutsumi, M., On coupled Klein-Gordon-Schrödinger equations II, J. Math. Anal. Appl., 66, 358-378 (1978) · Zbl 0396.35082 |
| [16] | Fukuda, I.; Tsutsumi, M., On coupled Klein-Gordon-Schrödinger equations III, Math. Jpn., 24, 307-321 (1979) · Zbl 0415.35071 |
| [17] | Guo, B. L., Global solution for some problem of a class of equations in interaction of complex Schrödinger field and real Klein-Gordon field, Sci. China Ser. A, 25, 97-107 (1982) |
| [18] | Guo, B. L.; Miao, C. X., Asymptotic behavior of coupled Klein-Gordon-Schrödinger equations, Sci. China Ser. A, 25, 705-714 (1995) |
| [19] | Guo, B. L.; Li, Y. S., Attractor for dissipative Klein-Gordon-Schrödinger equations in \(R^3\), J. Differ. Equat., 136, 356-377 (1997) · Zbl 0949.35021 |
| [20] | Hayashi, N.; von Wahl, W., On the global strong solutions of coupled Klein-Gordon-Schrödinger equations, J. Math. Soc. Jpn., 39, 489-497 (1987) · Zbl 0657.35034 |
| [21] | Hioe, F. T., Periodic solitary waves for two coupled nonlinear Klein-Gordon and Schrödinger equations, J. Phys. A: Math. Gen., 36, 7307-7330 (2003) · Zbl 1046.81019 |
| [22] | Huang, Z.; Jin, S.; Markowich, P. A.; Sparber, C.; Zheng, C., A time-splitting spectral scheme for the Maxwell-Dirac system, J. Comput. Phys., 208, 761-789 (2005) · Zbl 1070.81040 |
| [23] | Jin, S.; Markowich, P. A.; Zheng, C., Numerical simulation of a generalized Zakharov system, J. Comput. Phys., 201, 376-395 (2004) · Zbl 1079.65101 |
| [24] | Kong, L. H.; Liu, R. X.; Xu, Z. L., Numerical simulation of interaction between Schrödinger field and Klein-Gordon field by multisymplectic method, Appl. Math. Comp., 181, 342-350 (2006) · Zbl 1148.65317 |
| [25] | Li, Y.; Guo, B., Asymptotic smoothing effect of solutions to weakly dissiptive Klein-Gordon-Schrödinger equations, J. Math. Ann. Appl., 282, 256-265 (2003) · Zbl 1146.35321 |
| [26] | Liu, X.-Q.; Jiang, S., Exact solutions of multi-component nonlinear Schrödinger and Klein-Gordon equations, Appl. Math. Comp., 160, 875-880 (2005) · Zbl 1064.35192 |
| [27] | Liu, X.-S.; Qi, Y.-Y.; He, J.-F.; Ding, P. Z., Recent progress in sympletic algorithms for use in quantum systems, Commun. Comput. Phys., 2, 1-53 (2007) |
| [28] | Lu, K. N.; Wang, B. X., Global attractors for the Klein-Gordon-Schrödinger equation in unbounded domains, J. Differ. Equat., 170, 281-316 (2001) · Zbl 0979.35135 |
| [29] | Makhankov, V. G., Dynamics of classical solitons (in non-integrable systems), Phys. Lett. C, 35, 1-128 (1978) |
| [30] | Ohta, M., Stability of stationary states for the coupled Klein-Gordon-Schrödinger equations, Non. Anal., 27, 455-461 (1996) · Zbl 0867.35092 |
| [31] | Ozawa, T.; Tsutsumi, Y., Asymptotic behaviour of solutions for the coupled Klein-Gordon-Schrödinger equations, Adv. Stud. Pure Math., 23, 295-305 (1994) · Zbl 0817.35089 |
| [32] | Strang, G., On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5, 505-517 (1968) · Zbl 0184.38503 |
| [33] | Taha, T. R.; Ablowitz, M. J., Analytical and numerical aspects of certain nonlinear evolution equations,III. Numerical, nonlinear Schrödinger equation, J. Comput. Phys., 55, 203-230 (1984) · Zbl 0541.65082 |
| [34] | Veselic, K., On the nonrelativistic limit of the bound states of the Klein-Gordon equation, J. Math. Anal. Appl., 96, 63-84 (1983) · Zbl 0522.35008 |
| [35] | J. Wang, Multisymplectic Fourier pseudospectral method for coupled Klein-Gordon-Schrödinger equations, preprint.; J. Wang, Multisymplectic Fourier pseudospectral method for coupled Klein-Gordon-Schrödinger equations, preprint. |
| [36] | Wang, M. L.; Zhou, Y. B., The periodic wave solutions for the Klein-Gordon-Schrödinger equations, Phys. Lett. A, 318, 84-92 (2003) · Zbl 1098.81770 |
| [37] | Xiang, X. M., Spectral method for solving the system of equations of Schrödinger-Klein-Gordon field, J. Comput. Appl. Math., 21, 161-171 (1988) · Zbl 0634.65127 |
| [38] | Zhang, L. M., Convergence of a conservative difference scheme for a class of Klein-Gordon-Schrödinger equations in one space dimension, Appl. Math. Comput., 163, 343-355 (2005) · Zbl 1080.65084 |
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.
