A linearly implicit conservative scheme for the fractional nonlinear Schrödinger equation with wave operator

The present work is mainly devoted to studying the fractional nonlinear Schrödinger equation with wave operator. We first derive two conserved quantities of the equation, and then develop a three-level linearly implicit difference scheme. This scheme is shown to be conserves the discrete version of conserved quantities. Using energy method, we prove that the difference scheme is unconditionally stable, and the difference solution converges to the exact one with second order accuracy in both the space and time dimensions. Numerical experiments are performed to support our theoretical analysis and demonstrate the accuracy, discrete conservation laws and effectiveness for long-time simulation.

[1]  Weizhu Bao,et al.  Uniform Error Estimates of Finite Difference Methods for the Nonlinear Schrödinger Equation with Wave Operator , 2012, SIAM J. Numer. Anal..

[2]  Thierry Colin,et al.  Semidiscretization in time for nonlinear Schrödinger-waves equations , 1998 .

[3]  I. Turner,et al.  Numerical methods for fractional partial differential equations with Riesz space fractional derivatives , 2010 .

[4]  Manuel Duarte Ortigueira,et al.  Riesz potential operators and inverses via fractional centred derivatives , 2006, Int. J. Math. Math. Sci..

[5]  Jie Xin,et al.  The global solution for a class of systems of fractional nonlinear Schrödinger equations with periodic boundary condition , 2011, Comput. Math. Appl..

[6]  Wang Ting,et al.  Unconditional convergence of two conservative compact difference schemes for non-linear Schrdinger equation in one dimension , 2011 .

[7]  Wei Yang,et al.  Crank-Nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative , 2013, J. Comput. Phys..

[8]  Wei Yang,et al.  A linearly implicit conservative difference scheme for the space fractional coupled nonlinear Schrödinger equations , 2014, J. Comput. Phys..

[9]  H. Kober.  ON FRACTIONAL INTEGRALS AND DERIVATIVES , 1940 .

[10]  Luming Zhang,et al.  A compact finite difference scheme for the nonlinear Schrödinger equation with wave operator , 2012, Appl. Math. Comput..

[11]  S. Secchi,et al.  Soliton dynamics for fractional Schrödinger equations , 2013, 1305.1804.

[12]  Luming Zhang,et al.  A conservative numerical scheme for a class of nonlinear Schrödinger equation with wave operator , 2003, Appl. Math. Comput..

[13]  Zhi-Zhong Sun,et al.  On the L∞ convergence of a difference scheme for coupled nonlinear Schrödinger equations , 2010, Comput. Math. Appl..

[14]  L. Vu-Quoc,et al.  Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein-Gordon equation , 1995 .

[15]  Chengming Huang,et al.  An energy conservative difference scheme for the nonlinear fractional Schrödinger equations , 2015, J. Comput. Phys..

[16]  Jie Xin,et al.  Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation , 2008, Appl. Math. Comput..

[17]  S. Secchi.  Ground state solutions for nonlinear fractional Schrödinger equations in RN , 2012, 1208.2545.

[18]  Cem Çelik,et al.  Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative , 2012, J. Comput. Phys..

[19]  M. Meerschaert,et al.  Finite difference approximations for fractional advection-dispersion flow equations , 2004 .

[20]  Ting-chun Wang,et al.  Analysis of some new conservative schemes for nonlinear Schrödinger equation with wave operator , 2006, Appl. Math. Comput..

[21]  Fawang Liu,et al.  Numerical Methods for the Variable-Order Fractional Advection-Diffusion Equation with a Nonlinear Source Term , 2009, SIAM J. Numer. Anal..