An analysis of implicit conservative difference solver for fractional Klein-Gordon-Zakharov system

Abstract In this paper, we propose an efficient linearly implicit conservative difference solver for the fractional Klein–Gordon–Zakharov system. First of all, we present a detailed derivation of the energy conservation property of the system in the discrete setting. Then, by using the mathematical induction, it is proved that the proposed scheme is uniquely solvable. Subsequently, by virtue of the discrete energy method and a ‘cut-off’ function technique, it is shown that the proposed solver possesses the convergence rates of O ( Δ t 2 + h 2 ) in the sense of L∞- and L2- norms, respectively, and is unconditionally stable. Finally, numerical results testify the effectiveness of the proposed scheme and exhibit the correctness of theoretical results.

[1]  Changpin Li,et al.  Fractional-compact numerical algorithms for Riesz spatial fractional reaction-dispersion equations , 2016, 1608.03077.

[2]  Jiye Yang,et al.  Galerkin finite element method for two-dimensional Riesz space fractional diffusion equations , 2014, J. Comput. Phys..

[3]  Han Zhou,et al.  A class of second order difference approximations for solving space fractional diffusion equations , 2012, Math. Comput..

[4]  Anders Wäänänen,et al.  Advanced resource connector middleware for lightweight computational Grids , 2007 .

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

[6]  Weizhu Bao,et al.  Optimal error estimates of finite difference methods for the Gross-Pitaevskii equation with angular momentum rotation , 2012, Math. Comput..

[7]  Chengjian Zhang,et al.  Construction of high-order Runge-Kutta methods which preserve delay-dependent stability of DDEs , 2016, Appl. Math. Comput..

[8]  Chengjian Zhang,et al.  A new fourth-order numerical algorithm for a class of nonlinear wave equations , 2012 .

[9]  G. Akrivis,et al.  On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation , 1991 .

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

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

[12]  Jorge Eduardo Macías-Díaz,et al.  A structure-preserving method for a class of nonlinear dissipative wave equations with Riesz space-fractional derivatives , 2017, J. Comput. Phys..

[13]  Xuan Zhao,et al.  A Fourth-order Compact ADI scheme for Two-Dimensional Nonlinear Space Fractional Schrödinger Equation , 2014, SIAM J. Sci. Comput..

[14]  Chunmei Su,et al.  Uniform error bounds of a finite difference method for the Klein-Gordon-Zakharov system in the subsonic limit regime , 2016, Math. Comput..

[15]  Xinhua Zhang,et al.  Energy-preserving finite volume element method for the improved Boussinesq equation , 2014, J. Comput. Phys..

[16]  Jorge Eduardo Macías-Díaz,et al.  A pseudo energy-invariant method for relativistic wave equations with Riesz space-fractional derivatives , 2017, Comput. Phys. Commun..

[17]  K. Burrage,et al.  A new fractional finite volume method for solving the fractional diffusion equation , 2014 .

[18]  Weihua Deng,et al.  Polynomial spectral collocation method for space fractional advection–diffusion equation , 2012, 1212.3410.

[19]  Dong Liang,et al.  Energy-Conserved Splitting Finite-Difference Time-Domain Methods for Maxwell's Equations in Three Dimensions , 2010, SIAM J. Numer. Anal..

[20]  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..

[21]  Zuntao Fu,et al.  JACOBI ELLIPTIC FUNCTION EXPANSION METHOD AND PERIODIC WAVE SOLUTIONS OF NONLINEAR WAVE EQUATIONS , 2001 .

[22]  Bangti Jin,et al.  Error Analysis of a Finite Element Method for the Space-Fractional Parabolic Equation , 2014, SIAM J. Numer. Anal..

[23]  Chengjian Zhang,et al.  Analysis and application of a compact multistep ADI solver for a class of nonlinear viscous wave equations , 2015 .

[24]  Jian Wang,et al.  Solitary wave propagation and interactions for the Klein–Gordon–Zakharov equations in plasma physics , 2009 .

[25]  Fawang Liu,et al.  A Crank-Nicolson ADI Spectral Method for a Two-Dimensional Riesz Space Fractional Nonlinear Reaction-Diffusion Equation , 2014, SIAM J. Numer. Anal..

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

[27]  D. Furihata,et al.  Dissipative or Conservative Finite Difference Schemes for Complex-Valued Nonlinear Partial Different , 2001 .

[28]  Dong Liang,et al.  The time fourth-order compact ADI methods for solving two-dimensional nonlinear wave equations , 2018, Appl. Math. Comput..

[29]  Mehdi Dehghan,et al.  The solitary wave solution of coupled Klein-Gordon-Zakharov equations via two different numerical methods , 2013, Comput. Phys. Commun..

[30]  Wenwu Gao,et al.  An energy-momentum conserving scheme for Hamiltonian wave equation based on multiquadric trigonometric quasi-interpolation , 2018 .

[31]  Luigi Brugnano,et al.  Energy conservation issues in the numerical solution of the semilinear wave equation , 2014, Appl. Math. Comput..

[32]  Mark M. Meerschaert,et al.  A second-order accurate numerical approximation for the fractional diffusion equation , 2006, J. Comput. Phys..

[33]  Weizhu Bao,et al.  An Exponential Wave Integrator Sine Pseudospectral Method for the Klein-Gordon-Zakharov System , 2013, SIAM J. Sci. Comput..

[34]  Hong Wang,et al.  Wellposedness of Variable-Coefficient Conservative Fractional Elliptic Differential Equations , 2013, SIAM J. Numer. Anal..

[35]  Chengjian Zhang,et al.  A linearly implicit conservative scheme for the fractional nonlinear Schrödinger equation with wave operator , 2016, Int. J. Comput. Math..

[36]  Zhiyue Zhang,et al.  New energy-preserving schemes using Hamiltonian Boundary Value and Fourier pseudospectral methods for the numerical solution of the "good" Boussinesq equation , 2016, Comput. Phys. Commun..