Optimal error analysis of Crank-Nicolson schemes for a coupled nonlinear Schrödinger system in 3D

The paper is concerned with the time step condition of the commonly-used semi-implicit CrankNicolson finite difference schemes for a coupled nonlinear Schrdinger system in three dimensional space. We present the optimal L2 error estimate without any restriction on time step, while all previous works require certain time step conditions. Our approach is based on a rigorous analysis in both real and imaginary parts of the energy estimate (inequality) of the error function. Numerical examples for both two-dimensional and three-dimensional models are investigated and numerical results illustrate our theoretical analysis.

[1]  Stability of solitary waves for coupled nonlinear Schro¨dinger equations , 1996 .

[2]  Luming Zhang,et al.  New conservative difference schemes for a coupled nonlinear Schrödinger system , 2010, Appl. Math. Comput..

[3]  Zhi-Zhong Sun,et al.  The stability and convergence of a difference scheme for the Schrödinger equation on an infinite domain by using artificial boundary conditions , 2006, J. Comput. Phys..

[4]  Yu-Lin Chou.  Applications of Discrete Functional Analysis to the Finite Difference Method , 1991 .

[5]  Alfio Borzì,et al.  Analysis of a leap-frog pseudospectral scheme for the Schrödinger equation , 2006 .

[6]  Christophe Besse,et al.  Absorbing Boundary Conditions for General Nonlinear Schrödinger Equations , 2011, SIAM J. Sci. Comput..

[7]  Elder J. Villamizar-Roa,et al.  Self-similarity and asymptotic stability for coupled nonlinear Schrödinger equations in high dimensions , 2012 .

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

[9]  D. Griffiths,et al.  Introduction to Quantum Mechanics , 1960 .

[10]  Qianshun Chang,et al.  Difference Schemes for Solving the Generalized Nonlinear Schrödinger Equation , 1999 .

[11]  Weiwei Sun,et al.  Finite difference methods for a nonlinear and strongly coupled heat and moisture transport system in textile materials , 2012, Numerische Mathematik.

[12]  Lixin Wu,et al.  DuFort--Frankel-Type Methods for Linear and Nonlinear Schrödinger Equations , 1996 .

[13]  Y. Tourigny,et al.  Optimal H1 Estimates for two Time-discrete Galerkin Approximations of a Nonlinear Schrödinger Equation , 1991 .

[14]  Weiwei Sun,et al.  Error Estimates of Splitting Galerkin Methods for Heat and Sweat Transport in Textile Materials , 2013, SIAM J. Numer. Anal..

[15]  Ameneh Taleei,et al.  Numerical solution of nonlinear Schrödinger equation by using time‐space pseudo‐spectral method , 2010 .

[16]  J. M. Sanz-Serna,et al.  Methods for the numerical solution of the nonlinear Schroedinger equation , 1984 .

[17]  Georgios E. Zouraris,et al.  On the convergence of a linear two-step finite element method for the nonlinear Schrödinger equation , 2001 .

[18]  Zhi-Zhong Sun,et al.  Error Estimate of Fourth-Order Compact Scheme for Linear Schrödinger Equations , 2010, SIAM J. Numer. Anal..

[19]  C.-S. Chien,et al.  Multilevel Spectral-Galerkin and Continuation Methods for Nonlinear Schrödinger Equations , 2009, Multiscale Model. Simul..

[20]  Tingchun Wang,et al.  A robust semi-explicit difference scheme for the Kuramoto-Tsuzuki equation , 2009, J. Comput. Appl. Math..

[21]  Heping Ma,et al.  Error analysis for solving the Korteweg‐de Vries equation by a Legendre pseudo‐spectral method , 2000 .

[22]  Sergey Leble,et al.  On convergence and stability of a numerical scheme of Coupled Nonlinear Schrödinger Equations , 2008, Comput. Math. Appl..

[23]  Qianshun Chang,et al.  Finite difference method for generalized Zakharov equations , 1995 .

[24]  M. Salazar-Palma,et al.  An Unconditionally Stable Scheme for the , 2003 .

[25]  Thiab R. Taha,et al.  Numerical simulation of coupled nonlinear Schrödinger equation , 2001 .

[26]  Yinnian He,et al.  Finite element approximation for the viscoelastic fluid motion problem , 2003 .

[27]  Alice C. Yew,et al.  Stability Analysis of Multipulses in Nonlinearly-Coupled Schrodinger Equations , 2000 .

[28]  Graeme Fairweather,et al.  Discrete-time Orthogonal Spline Collocation Methods for Schrödinger Equations in Two Space Variables , 1998 .

[29]  A. G. Bratsos.  A modified numerical scheme for the cubic Schrödinger equation , 2011 .

[30]  Thiab R. Taha,et al.  A linearly implicit conservative scheme for the coupled nonlinear Schrödinger equation , 2007, Math. Comput. Simul..

[31]  Tony F. Chan,et al.  Stability analysis of difference schemes for variable coefficient Schro¨dinger type equations , 1987 .

[32]  Graeme Fairweather,et al.  Three level Galerkin methods for parabolic equations , 1974 .

[33]  M. Radziunas,et al.  On Convergence and Stability of the Explicit Difference Method for Solution of Nonlinear Schrödinger Equations , 1999 .

[34]  R. Glassey.  Convergence of an energy-preserving scheme for the Zakharov equations in one space dimension , 1992 .

[35]  Guangwei Yuan,et al.  Convergence and stability of explicit/implicit schemes for parabolic equations with discontinuous coefficients , 2004 .

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

[37]  Akira Hasegawa,et al.  Optical solitons in fibers , 1993, International Commission for Optics.

[38]  C. Menyuk,et al.  Stability of solitons in birefringent optical fibers. I: equal propagation amplitudes. , 1987, Optics letters.