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New accurate conservative finite difference schemes for 1-D and 2-D Schrödinger-Boussinesq Equations

Year 2024, Volume: 45 Issue: 4, 777 - 788, 30.12.2024

Ayhan Aydın , Taha Mohammed

https://doi.org/10.17776/csj.1445761

Abstract

In this paper, first-order and second-order accurate structure-preserving finite difference schemes are proposed for solving the Schrödinger- Boussinesq equations. The conservation of the discrete energy and mass of the present schemes are analytically proved. Numerical experiments are given to support the theoretical results. Numerical examples show the efficiency of the proposed scheme and the correction of the theoretical proofs

Keywords

Schrödinger- Boussinesq equations conservative numerical methods partitioned average vector field method soliton solution

References

  • [1] Zhang H., Song S.H., Chen X.D., Zhou W.E., Average vector field methods for the coupled Schrödinger—KdV equations, Chinese Physics B, 23(7) (2014) 070208.
  • [2] Aydın A., Multisymplectic integration of N-coupled nonlinear Schrödinger equation with destabilized periodic wave solutions, Chaos, Solitons & Fractals, 41(2) (2009) 735-751.
  • [3] Wang L., Wang Y., Multisymplectic structure-preserving scheme for the coupled Gross–Pitaevskii equations, International Journal of Computer Mathematics, 98(4) (2021) 783-806.
  • [4] Yajima N., Satsuma J., Soliton solutions in a diatomic lattice system, Progress of Theoretical Physics, 62(2) (1979) 370-378.
  • [5]Rao N.N., Coupled scalar field equations for nonlinear wave modulations in dispersive media, Pramana, 46 (1996) 161-202.
  • [6] Huang L.Y., Jiao Y.D., Liang D.M., Multi-symplectic scheme for the coupled Schrödinger—Boussinesq equations, Chinese Physics B, 22(7) (2013) 070201.
  • [7] Bai D., Zhang L., The quadratic B-spline finite-element method for the coupled Schrödinger–Boussinesq equations, International Journal of Computer Mathematics, 88(8) (2011) 1714-1729.
  • [8] Zhang L., Bai D., Wang S., Numerical analysis for a conservative difference scheme to solve the Schrödinger–Boussinesq equation, Journal of computational and applied mathematics, 235(17) (2011) 4899-4915.
  • [9] Fei Z., Pérez-García V.M., Vázquez L., Numerical simulation of nonlinear Schrödinger systems: a new conservative scheme, Applied Mathematics and Computation, 71(2-3) (1995) 165-177.
  • [10] Bai D., Wang J., The time-splitting Fourier spectral method for the coupled Schrödinger–Boussinesq equations. Communications in Nonlinear Science and Numerical Simulation, 17(3) (2012) 1201-1210.
  • [11] Liao F., Zhang L., Wang S., Time-splitting combined with exponential wave integrator Fourier pseudospectral method for Schrödinger–Boussinesq system, Communications in Nonlinear Science and Numerical Simulation, 55 (2018) 93-104.
  • [12] Hairer E., Wanner G., Lubich C., Hairer E., Wanner G., Lubich C., Symmetric Integration and Reversibility, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, (2006) 143-178.
  • [13] Quispel G.R.W., McLaren D.I., A new class of energy-preserving numerical integration methods, Journal of Physics A: Mathematical and Theoretical, 41(4) (2008) 045206.
  • [14] Celledoni E., Grimm V., McLachlan R.I., McLaren D.I., O’Neale D., Owren B. and Quispel G.R.W., Preserving energy resp. dissipation in numerical PDEs using the “Average Vector Field” method, Journal of Computational Physics, 231(20) (2012) 6770-6789. [15] Wang L., Cai W., Wang Y., An energy-preserving scheme for the coupled Gross-Pitaevskii equations, Adv. Appl. Math. Mech., 13(1) (2021) 203-231.
  • [16] Cai W., Li H., Wang Y., Partitioned averaged vector field methods. Journal of Computational Physics, 370 (2018) 25-42.
  • [17] Wang J., Conservative Fourier spectral scheme for the coupled Schrödinger–Boussinesq equations, Advances in Difference Equations, (2018) 1-19.
  • [18] Aydin A, Mohammed T.Y., An Energy Preserving Scheme for CSB System, 2^nd Int. Grad. Stud.Cong. (IGSCONG-22), Proceeding Book, ISBN: 978-605-73639-2-3, 91-101.
  • [19] Cai J., Chen J., Yang B., Efficient energy-preserving wavelet collocation schemes for the coupled nonlinear Schrödinger-Boussinesq system, Applied Mathematics and Computation, 357 (2019) 1-11.
  • [20] Liao F., Zhang L., Conservative compact finite difference scheme for the coupled S chrödinger–B oussinesq equation, Numerical Methods for Partial Differential Equations, 32(6) (2016) 1667-1688.
  • [21] Aydın A., Karasözen B., Lobatto IIIA–IIIB discretization of the strongly coupled nonlinear Schrödinger equation, Journal of computational and applied mathematics, 235(16) (2011) 4770-4779.
  • [22] Aydın A., Karasözen B., Symplectic and multi-symplectic methods for coupled nonlinear Schrödinger equations with periodic solutions, Computer Physics Communications, 177(7) (2007) 566-583.
  • [23] Leimkuhler B., Reich S., Simulating Hamiltonian dynamics (No. 14). Cambridge University Press, (2004)
  • [24] Aydin, A., Sabawe, BAK., New conservative schemes for Zakharov equation, Turk. J. Math. Comput. Sci. 15(2) (2023) 227-293.
  • [25] Ismail M. S., Ashi H. A., A compact finite difference scheme for solving the coupled nonlinear Schrodinger-Boussinesq equations, Applied Mathematics , 7 (2016) 605-615.
  • [26] Hu X., Wang S., Zhang L., Maximum error estimates for a compact difference scheme of the coupled nonlinear Schrödinger–Boussinesq equations, Numerical Methods for Partial Differential Equations, 35(6) (2019) 1971-1999.
  • [27] Deng D., Wu Q., Analysis of the linearly energy-and mass-preserving finite difference methods for the coupled Schrödinger-Boussinesq equations, Applied Numerical Mathematics, 170 (2021) 14-38.
  • [28] Tingchun W., Boling G., Qiubin X., Fourth-order compact and energy conservative difference schemes for the nonlinear Schrödinger equation in two dimensions, Journal of Computational Physics, 243 (2013) 382–399.
  • [29] Hong-Lin L., Zhi-Zhong S., Han-Sheng S., Error Estimate of Fourth-Order Compact Scheme For Linear Schrödinger Equations, SIAM J. Numer. Anal., 47(6) (2010) 4381–4401.
  • [30] Gholamreza K., Hadi MF., The Matrix Transformation Technique for the Time-Space Fractional Linear Schrödinger Equation, Iranian Journal of Mathematical Chemistry, 15(3) (2024) 137-154.
  • [31] Gholamreza K., Hadi MF., Unconditionally stable finite element method for the variable-order fractional Schrödinger equation with Mittag-Leffler kernel, Journal of Mathematical Modeling, 12(3) (2024) 533–550.

Year 2024, Volume: 45 Issue: 4, 777 - 788, 30.12.2024

Ayhan Aydın , Taha Mohammed

https://doi.org/10.17776/csj.1445761

Abstract

References

  • [1] Zhang H., Song S.H., Chen X.D., Zhou W.E., Average vector field methods for the coupled Schrödinger—KdV equations, Chinese Physics B, 23(7) (2014) 070208.
  • [2] Aydın A., Multisymplectic integration of N-coupled nonlinear Schrödinger equation with destabilized periodic wave solutions, Chaos, Solitons & Fractals, 41(2) (2009) 735-751.
  • [3] Wang L., Wang Y., Multisymplectic structure-preserving scheme for the coupled Gross–Pitaevskii equations, International Journal of Computer Mathematics, 98(4) (2021) 783-806.
  • [4] Yajima N., Satsuma J., Soliton solutions in a diatomic lattice system, Progress of Theoretical Physics, 62(2) (1979) 370-378.
  • [5]Rao N.N., Coupled scalar field equations for nonlinear wave modulations in dispersive media, Pramana, 46 (1996) 161-202.
  • [6] Huang L.Y., Jiao Y.D., Liang D.M., Multi-symplectic scheme for the coupled Schrödinger—Boussinesq equations, Chinese Physics B, 22(7) (2013) 070201.
  • [7] Bai D., Zhang L., The quadratic B-spline finite-element method for the coupled Schrödinger–Boussinesq equations, International Journal of Computer Mathematics, 88(8) (2011) 1714-1729.
  • [8] Zhang L., Bai D., Wang S., Numerical analysis for a conservative difference scheme to solve the Schrödinger–Boussinesq equation, Journal of computational and applied mathematics, 235(17) (2011) 4899-4915.
  • [9] Fei Z., Pérez-García V.M., Vázquez L., Numerical simulation of nonlinear Schrödinger systems: a new conservative scheme, Applied Mathematics and Computation, 71(2-3) (1995) 165-177.
  • [10] Bai D., Wang J., The time-splitting Fourier spectral method for the coupled Schrödinger–Boussinesq equations. Communications in Nonlinear Science and Numerical Simulation, 17(3) (2012) 1201-1210.
  • [11] Liao F., Zhang L., Wang S., Time-splitting combined with exponential wave integrator Fourier pseudospectral method for Schrödinger–Boussinesq system, Communications in Nonlinear Science and Numerical Simulation, 55 (2018) 93-104.
  • [12] Hairer E., Wanner G., Lubich C., Hairer E., Wanner G., Lubich C., Symmetric Integration and Reversibility, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, (2006) 143-178.
  • [13] Quispel G.R.W., McLaren D.I., A new class of energy-preserving numerical integration methods, Journal of Physics A: Mathematical and Theoretical, 41(4) (2008) 045206.
  • [14] Celledoni E., Grimm V., McLachlan R.I., McLaren D.I., O’Neale D., Owren B. and Quispel G.R.W., Preserving energy resp. dissipation in numerical PDEs using the “Average Vector Field” method, Journal of Computational Physics, 231(20) (2012) 6770-6789. [15] Wang L., Cai W., Wang Y., An energy-preserving scheme for the coupled Gross-Pitaevskii equations, Adv. Appl. Math. Mech., 13(1) (2021) 203-231.
  • [16] Cai W., Li H., Wang Y., Partitioned averaged vector field methods. Journal of Computational Physics, 370 (2018) 25-42.
  • [17] Wang J., Conservative Fourier spectral scheme for the coupled Schrödinger–Boussinesq equations, Advances in Difference Equations, (2018) 1-19.
  • [18] Aydin A, Mohammed T.Y., An Energy Preserving Scheme for CSB System, 2^nd Int. Grad. Stud.Cong. (IGSCONG-22), Proceeding Book, ISBN: 978-605-73639-2-3, 91-101.
  • [19] Cai J., Chen J., Yang B., Efficient energy-preserving wavelet collocation schemes for the coupled nonlinear Schrödinger-Boussinesq system, Applied Mathematics and Computation, 357 (2019) 1-11.
  • [20] Liao F., Zhang L., Conservative compact finite difference scheme for the coupled S chrödinger–B oussinesq equation, Numerical Methods for Partial Differential Equations, 32(6) (2016) 1667-1688.
  • [21] Aydın A., Karasözen B., Lobatto IIIA–IIIB discretization of the strongly coupled nonlinear Schrödinger equation, Journal of computational and applied mathematics, 235(16) (2011) 4770-4779.
  • [22] Aydın A., Karasözen B., Symplectic and multi-symplectic methods for coupled nonlinear Schrödinger equations with periodic solutions, Computer Physics Communications, 177(7) (2007) 566-583.
  • [23] Leimkuhler B., Reich S., Simulating Hamiltonian dynamics (No. 14). Cambridge University Press, (2004)
  • [24] Aydin, A., Sabawe, BAK., New conservative schemes for Zakharov equation, Turk. J. Math. Comput. Sci. 15(2) (2023) 227-293.
  • [25] Ismail M. S., Ashi H. A., A compact finite difference scheme for solving the coupled nonlinear Schrodinger-Boussinesq equations, Applied Mathematics , 7 (2016) 605-615.
  • [26] Hu X., Wang S., Zhang L., Maximum error estimates for a compact difference scheme of the coupled nonlinear Schrödinger–Boussinesq equations, Numerical Methods for Partial Differential Equations, 35(6) (2019) 1971-1999.
  • [27] Deng D., Wu Q., Analysis of the linearly energy-and mass-preserving finite difference methods for the coupled Schrödinger-Boussinesq equations, Applied Numerical Mathematics, 170 (2021) 14-38.
  • [28] Tingchun W., Boling G., Qiubin X., Fourth-order compact and energy conservative difference schemes for the nonlinear Schrödinger equation in two dimensions, Journal of Computational Physics, 243 (2013) 382–399.
  • [29] Hong-Lin L., Zhi-Zhong S., Han-Sheng S., Error Estimate of Fourth-Order Compact Scheme For Linear Schrödinger Equations, SIAM J. Numer. Anal., 47(6) (2010) 4381–4401.
  • [30] Gholamreza K., Hadi MF., The Matrix Transformation Technique for the Time-Space Fractional Linear Schrödinger Equation, Iranian Journal of Mathematical Chemistry, 15(3) (2024) 137-154.
  • [31] Gholamreza K., Hadi MF., Unconditionally stable finite element method for the variable-order fractional Schrödinger equation with Mittag-Leffler kernel, Journal of Mathematical Modeling, 12(3) (2024) 533–550.
There are 30 citations in total.

Details

Primary Language English
Subjects Numerical Solution of Differential and Integral Equations
Journal Section Natural Sciences
Authors

Ayhan Aydın 0000-0002-0837-9364

Taha Mohammed 0000-0002-5679-9195

Publication Date December 30, 2024
Submission Date March 2, 2024
Acceptance Date December 12, 2024
Published in Issue Year 2024Volume: 45 Issue: 4

Cite

APA Aydın, A., & Mohammed, T. (2024). New accurate conservative finite difference schemes for 1-D and 2-D Schrödinger-Boussinesq Equations. Cumhuriyet Science Journal, 45(4), 777-788. https://doi.org/10.17776/csj.1445761
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