Numerical analysis of cubic orthogonal spline collocation methods for the coupled Schrödinger-Boussinesq equations. (English) Zbl 1368.65199
Summary: In this article, we formulate two orthogonal spline collocation schemes, which consist of a nonlinear and a linear scheme for solving the coupled Schrödinger-Boussinesq equations numerically. Firstly, the conservation laws of our schemes are derived. Secondly, the existence solutions of our schemes are investigated. Thirdly, the convergence and stability of the nonlinear scheme are analyzed by means of discrete energy methods, while the convergence of the linear scheme is proved by cut-off function technique. Finally, numerical results are reported to verify our theoretical analysis for the numerical methods.
MSC:
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
orthogonal spline collocation; Schrödinger-Boussinesq equations; conservation law; convergence; stability; numerical resultReferences:
| [1] | Amodio, P.; Cash, J. R.; Roussos, G.; Wright, R. W., Almost block diagonal linear systems: sequential and parallel solution techniques, and applications, Numer. Linear Algebra Appl., 7, 275-317 (2000) · Zbl 1051.65018 |
| [2] | Bai, D. M.; Zhang, L. M., The quadratic B-spline finite element method for the coupled Schrödinger-Boussinesq equations, Int. J. Comput. Math., 88, 1714-1729 (2011) · Zbl 1216.35130 |
| [3] | Bao, W. Z.; Cai, Y. Y., Uniform error estimates of finite difference methods for the nonlinear Schrödinger equation with wave operator, SIAM J. Numer. Anal., 50, 492-521 (2012) · Zbl 1246.35188 |
| [4] | Bao, W. Z.; Cai, Y. Y., Optimal error estimates of finite difference methods for the Gross-Pitaevskii equation with angular momentum rotation, Math. Comput., 281, 99-128 (2013) · Zbl 1264.65146 |
| [5] | Bialecki, B.; Fairweather, G., Orthogonal spline collocation methods for partial differential equations, J. Comput. Appl. Math., 128, 55-82 (2001) · Zbl 0971.65105 |
| [6] | Bilige, S.; Chaolu, T.; Wang, X. M., Application of the extended simplest equation method to the coupled Schrödinger-Boussinesq equation, Appl. Math. Comput., 224, 517-523 (2013) · Zbl 1334.35308 |
| [7] | Boor, C. D.; Weiss, R., Algorithm 546: SOLVEBLOK: a package for solving almost block diagonal linear systems, ACM Trans. Math. Softw., 6, 80-91 (1980) · Zbl 0434.65013 |
| [8] | Ciarlet, P. G., The Finite Element Method for Elliptic Problems (1978), North-Holland: North-Holland Amsterdam · Zbl 0445.73043 |
| [9] | Danumjaya, P.; Pani, A. K., Orthogonal cubic spline collocation method for the extended Fisher-Kolmogorov equation, J. Comput. Appl. Math., 174, 101-117 (2005) · Zbl 1067.65107 |
| [10] | Diaz, J. C.; Fairweather, G.; Keast, P., FORTRAN packages for solving certain almost block diagonal linear systems by modified alternate row and column elimination, ACM Trans. Math. Softw., 9, 358-375 (1983) · Zbl 0516.65013 |
| [11] | Diaz, J. C.; Fairweather, G.; Keast, P., Algorithm 603 COLROW and ARCECO: FORTRAN packages for solving certain almost block diagonal linear systems by modified alternate row and column elimination, ACM Trans. Math. Softw., 9, 376-380 (1983) · Zbl 0516.65013 |
| [12] | Fairweather, G.; Gladwell, I., Algorithms for almost block diagonal linear systems, SIAM Rev., 46, 49-58 (2004) · Zbl 1062.65031 |
| [13] | Fairweather, G.; Yang, X. H.; Da, X.; Zhang, H. X., An ADI Crank-Nicolson orthogonal spline collocation method for the two-dimensional fractional diffusion-wave equation, J. Sci. Comput., 65, 1217-1239 (2015) · Zbl 1328.65216 |
| [14] | Farah, L. G.; Pastor, A., On the periodic Schrödinger-Boussinesq system, J. Math. Anal. Appl., 368, 330-349 (2010) · Zbl 1190.35208 |
| [15] | Fernandes, R. I.; Bialecki, B.; Fairweather, G., An ADI extrapolated Crank-Nicolson orthogonal spline collocation method for nonlinear reaction-diffusion systems on evolving domains, J. Comput. Phys., 299, 561-580 (2015) · Zbl 1352.65393 |
| [16] | Fernandes, R. I.; Fairweather, G., An ADI extrapolated Crank-Nicolson orthogonal spline collocation method for nonlinear reaction-diffusion systems, J. Comput. Phys., 231, 6248-6267 (2012) · Zbl 1284.65026 |
| [17] | Greenwell-Yanik, C. E.; Fairweather, G., Analysis of spline collocation methods for parabolic and hyperbolic problems in two space variables, SIAM J. Numer. Anal., 23, 282-296 (1986) · Zbl 0595.65122 |
| [18] | Guo, B. L., The global solution of the system of equations for complex Schrödinger field coupled with Boussinesq type self-consistent field, Acta Math. Sin., 26, 295-306 (1983), (in Chinese) · Zbl 0554.35102 |
| [19] | Guo, B. L.; Chen, G. N., The convergence of Galerkin-Fourier method for equation of Schrödinger-Boussinesq field, J. Comput. Math., 2, 344-355 (1984) · Zbl 0565.65077 |
| [20] | Guo, B. L.; Chen, F. X., Finite dimensional behavior of global attractors for weakly damped nonlinear Schrödinger-Boussinesq equations, Physica D, 93, 101-118 (1996) · Zbl 0885.35123 |
| [21] | Guo, B. L.; Du, X. Y., The behavior of attractors for damped Schrödinger-Boussinesq equation, Commun. Nonlinear Sci. Numer. Simul., 6, 54-60 (2001) · Zbl 0976.35081 |
| [22] | Guo, B. L.; Du, X. Y., Existence of the periodic solution for the weakly damped Schrödinger-Boussinesq equation, J. Math. Anal. Appl., 262, 453-472 (2001) · Zbl 1040.35114 |
| [23] | Hon, Y. V.; Fan, E. G., A series of exact solutions for coupled Higgs field equation and coupled Schrödinger-Boussinesq equation, Nonlinear Anal., 71, 3501-3508 (2009) · Zbl 1172.35480 |
| [24] | Huang, L. Y.; Jiao, Y. D.; Liang, D. M., Multi-symplectic scheme for the coupled Schrödinger-Boussinesq equations, Chin. Phys. B, 7, 1-5 (2013) |
| [25] | Khebchareon, M.; Pani, A. K.; Faiweather, G., Convergence analysis of Crank-Nicolson orthogonal spline collocation methods for linear parabolic problems in two space variables, Int. J. Numer. Anal. Model., 13, 58-72 (2016) · Zbl 1347.65147 |
| [26] | Kilicman, A.; Abazari, R., Travelling wave solutions of the Schrödinger-Boussinesq system, Abstr. Appl. Anal., 2012, 1-11 (2012) · Zbl 1253.65162 |
| [27] | Li, Y. S.; Chen, Q. Y., Finite dimensional global attractor for dissipative Schrödinger-Boussinesq equations, J. Math. Anal. Appl., 205, 107-132 (1997) · Zbl 0958.35129 |
| [28] | Li, B.; Fairweather, G.; Bialecki, B., Discrete-time orthogonal spline collocation methods for Schrödinger equations in two space variables, SIAM J. Numer. Anal., 35, 453-477 (1998) · Zbl 0911.65095 |
| [29] | Li, C.; Zhao, T. G.; Deng, W. H.; Wu, Y. J., Orthogonal spline collocation methods for the subdiffusion equation, J. Comput. Appl. Math., 255, 517-528 (2014) · Zbl 1291.65307 |
| [30] | Liao, F.; Zhang, L. M., Conservative compact finite difference scheme for the coupled Schrödinger-Boussinesq equation, Numer. Methods Partial Differ. Equ., 32, 1667-1688 (2016) · Zbl 1360.65213 |
| [31] | Manickam, A. V.; Moudgalya, K. M.; Pani, A. K., Second-order splitting combined with orthogonal cubic spline collocation method for the Kuramoto-Sivashinsky equation, Comput. Math. Appl., 35, 5-25 (1998) · Zbl 0907.65098 |
| [32] | Meng, Q. J.; Yin, L. P.; Jin, X. Q.; Qiao, F. L., Numerical solutions of the coupled nonlinear Schrödinger equations by orthogonal spline collocation method, Commun. Comput. Phys., 12, 1392-1416 (2012) · Zbl 1388.65165 |
| [33] | Rao, N. N., Exact solutions of coupled scalar field equations, J. Phys. A, Math. Gen., 22, 4813-4825 (1989) · Zbl 0718.34002 |
| [34] | Rao, N. N., Coupled scalar field equations for nonlinear wave modulations in dispersive media, Pramana J. Phys., 46, 161-202 (1991) |
| [35] | Sun, Z. Z., A note on finite difference method for generalized Zakharov equations, J. Southeast Univ., 16, 84-86 (2000) · Zbl 1007.65052 |
| [36] | Wang, T. C.; Guo, B. L.; Zhang, L. M., New conservative difference schemes for a coupled nonlinear Schrödinger system, Appl. Math. Comput., 217, 1604-1619 (2010) · Zbl 1205.65242 |
| [37] | Wang, S. S.; Zhang, L. M., A class of conservative orthogonal spline collocation schemes for solving coupled Klein-Gordin-Schrödinger equations, Appl. Math. Comput., 203, 799-812 (2008) · Zbl 1180.65135 |
| [38] | Wang, S. S.; Zhang, L. M., 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 |
| [39] | Wang, T. C.; Zhao, X. F., Optimal \(l^\infty\) error estimates of finite difference methods for the coupled Gross-Pitaevskii equations in high dimensions, Sci. China Math., 57, 2189-2214 (2014) · Zbl 1320.65130 |
| [40] | Xia, Y. R.; Bin, L. Z., Exact explicit solutions of the nonlinear Schrödinger equation coupled to the Boussinesq equation, Acta Math. Sci., 23B, 453-460 (2003) · Zbl 1038.35129 |
| [41] | Yajima, N.; Satsuma, J., Soliton solutions in a diatomic lattice system, Prog. Theor. Phys., 62, 370-378 (1979) |
| [42] | Zhang, L. M.; Bai, D. M.; Wang, S. S., Numerical analysis for a conservative difference scheme to solve the Schrödinger-Boussinesq equation, J. Comput. Appl. Math., 235, 4899-4915 (2011) · Zbl 1230.65099 |
| [43] | Zhao, X. F., An exponential wave integrator pseudospectral method for the symmetric regularized-long-wave equation, J. Comput. Math., 34, 49-69 (2016) · Zbl 1363.65179 |
| [44] | Zheng, J. D.; Xiang, X. M., The finite element analysis for the equation system coupling the complex Schrödinger and real Boussinesq fields, Math. Numer. Sin., 2, 344-355 (1984), (in Chinese) · Zbl 0565.65077 |
| [45] | Zhou, Y., Application of Discrete Functional Analysis to the Finite Difference Methods (1990), International Academic Publishers: International Academic Publishers Beijing |
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