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A Conservative Local Discontinuous Galerkin Method for the Schrödinger-KdV System

Published online by Cambridge University Press:  03 June 2015

Yinhua Xia  and
Yan Xu
Yinhua Xia*
Affiliation:
School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, P.R. China
Yan Xu*
Affiliation:
School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, P.R. China
*
Corresponding author.Email:[email protected]
Email:[email protected]
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Abstract

In this paper we develop a conservative local discontinuous Galerkin (LDG) method for the Schrödinger-Korteweg-de Vries (Sch-KdV) system, which arises in various physical contexts as a model for the interaction of long and short nonlinear waves. Conservative quantities in the discrete version of the number of plasmons, energy of the oscillations and the number of particles are proved for the LDG scheme of the Sch-KdV system. Semi-implicit time discretization is adopted to relax the time step constraint from the high order spatial derivatives. Numerical results for accuracy tests of stationary traveling soliton, and the collision of solitons are shown. Numerical experiments illustrate the accuracy and capability of the method.

Keywords

65M6035Q5335Q55Schrödinger-KdV systemthe conservative local discontinuous Galerkin methodsemi-implicit time discretizationconservative quantities
Type
Research Article
Information
Communications in Computational Physics , Volume 15 , Issue 4 , April 2014 , pp. 1091 - 1107
Copyright
Copyright © Global Science Press Limited 2014

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References

[1]Albert, J. and Angulo Pava, J., Existence and stability of ground-state solutions of a Schrödinger-KdV system, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 987–1029.CrossRefGoogle Scholar
[2]Appert, K. and Vaclavik, J., Dynamics of coupled solitons, Phys. Fluids, 20 (1977), 1845–1849.Google Scholar
[3]Bai, D.M. and Zhang, L.M., The finite element method for the coupled Schrödinger-KdV equations, Phys. Lett. A, 373 (2009), 2237–2244.Google Scholar
[4]Bai, D.M. and Zhang, L.M., Numerical studies on a novel split-step quadratic B-spline finite element method for the coupled Schrödinger-KdV equations, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1263–1273.Google Scholar
[5]Benney, D.J., A general theory for interactions between short and long waves, Stud. Appl. Math., 56 (1977), 81–94.Google Scholar
[6]Bona, J.L., Chen, H., Karakashian, O. and Xing, Y., Conservative, discontinuous-Galerkin methods for the generalized Korteweg-de Vries equation, Math. Comp., 82(2013), 1401–1432.Google Scholar
[7]Cockburn, B., Hou, S. and Shu, C.-W., The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case, Math. Comp., 54 (1990), 545–581.Google Scholar
[8]Cockburn, B., Lin, S.-Y. and Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one dimensional systems, J. Comput. Phys., 84 (1989), 90–113.Google Scholar
[9]Cockburn, B. and Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework, Math. Comp., 52 (1989), 411–435.Google Scholar
[10]Cockburn, B. and Shu, C.-W., The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, J. Comput. Phys., 141 (1998), 199–224.CrossRefGoogle Scholar
[11]Cockburn, B. and Shu, C.-W., The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal., 35 (1998), 2440–2463.CrossRefGoogle Scholar
[12]Golbabai, A. and Safdari-Vaighani, A., A meshless method for numerical solution of the coupled Schrödinger-KdV equations, Computing, 92 (2011), 225–242.Google Scholar
[13]Kawahara, T., Sugimoto, N. and Kakutani, T., Nonlinear interaction between short and long capillary-gravity waves, J. Phys. Soc. Japan, 39 (1975), 1379–1386.Google Scholar
[14]Makhankov, V., On stationary solutions of the Schrödinger equation with a self-consistent potential satisfying Boussinesq’s equation, Phys. Lett. A, 50 (1974), 42–44.Google Scholar
[15]Nishikawa, K., Hojo, H., Mima, K. and Ikezi, H., Coupled nonlinear electron-plasma and ion-acoustic waves, Phys. Rev. Lett., 33 (1974), 148–151.Google Scholar
[16]Reed, W.H. and Hill, T.R., Triangular mesh method for the neutron transport equation, Technical report LA-UR-73-479, Los Alamos Scientific Laboratory, Los Alamos, NM, 1973.Google Scholar
[17]Xia, Y., Xu, Y. and Shu, C.-W., Efficient time discretization for local discontinuous Galerkin methods, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 677–693.Google Scholar
[18]Xia, Y., Xu, Y. and Shu, C.-W., Local discontinuous Galerkin methods for the generalized Zakharov system, J. Comput. Phys. 229 (2010), 1238–1259.Google Scholar
[19]Xu, Y. and Shu, C.-W., Local discontinuous Galerkin methods for nonlinear Schrodinger equations, J. Comput. Phys., 205 (2005), 72–97.Google Scholar
[20]Xu, Y. and Shu, C.-W., Local discontinuous Galerkin methods for high-order time-dependent partial differential equations, Communications in Computational Physics, 7 (2010), 1–46.Google Scholar
[21]Yajima, N. and Satsuma, J., Soliton solutions in a diatomic lattice system, Prog. Theor. Phys., 62 (1979), 370–378.Google Scholar
[22]Yan, J. and Shu, C.-W., Local discontinuous Galerkin methods for partial differential equations with higher order derivatives, J. Sci. Comput., 17 (2002), 27–47.Google Scholar