The studies of the linearly modified energy-preserving finite difference methods applied to solve two-dimensional nonlinear coupled wave equations

In this paper, four linearly energy-preserving finite difference methods (EP-FDMs) are designed for two-dimensional (2D) nonlinear coupled sine-Gordon equations (CSGEs) and coupled Klein-Gordon equations (CKGEs) using the invariant energy quadratization method (IEQM). The 1st EP-FDM is designed by first introducing two auxiliary functions to rewrite the original problems into the new system only including the 1st-order temporal derivatives, and then applying Crank-Nicolson (C-N) method and 2nd-order centered difference methods for the discretizations of temporal and spatial derivatives, respectively. The 2nd EP-FDM is directly devised based on the uses of 2nd-order centered difference methods to approximate 2nd-order temporal and spatial derivatives. The 1st and 2nd EP-FDMs need numerical solutions of the algebraic system with variable coefficient matrices at each time level. By modifying the 2nd EP-FDM, the 3rd EP-FDM, which is implemented by computing the system of algebraic equations with constant coefficient matrices at each time level, is developed. Finally, an energy-preserving alternating direction implicit (ADI) finite difference method (EP-ADI-FDM) is established by a combination of ADI method with the 3rd EP-FDM. By using the discrete energy method, it is shown that they are all uniquely solvable, and their solutions have a convergent rate of ${\mathscr{O}}(\varDelta t^{2}+{h^{2}_{x}}+{h^{2}_{y}})$ in H1-norm and satisfy the discrete conservative laws. Numerical results show the efficiency and accuracy of them.

[1]  V. Antony Vijesh,et al.  Chebyshev Wavelet Quasilinearization Scheme for Coupled Nonlinear Sine-Gordon Equations , 2017 .

[2]  Mehdi Dehghan,et al.  The use of radial basis functions (RBFs) collocation and RBF-QR methods for solving the coupled nonlinear sine-Gordon equations , 2015 .

[3]  S. Arabia,et al.  CONSERVATION LAWS OF COUPLED KLEIN-GORDON EQUATIONS WITH CUBIC AND POWER LAW NONLINEARITIES , 2014 .

[4]  W. Cai,et al.  Partitioned averaged vector field methods , 2017, J. Comput. Phys..

[5]  Hiroshi Sugiura,et al.  Regions of convergence and dynamics of Schröder-like iteration formulae as applied to complex polynomial equations with multiple roots , 2019, Numerical Algorithms.

[6]  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..

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

[8]  B. Guo,et al.  Analysis of some finite difference schemes for two‐dimensional Ginzburg‐Landau equation , 2011 .

[9]  Wei Liu,et al.  A finite difference approximation of reduced coupled model for slightly compressible Forchheimer fractures in Karst aquifer system , 2019, Numerical Algorithms.

[10]  Doris Kuhlmann,et al.  On the Theory of Plastic Deformation , 1951 .

[11]  Chengjian Zhang,et al.  Preconditioned quasi-compact boundary value methods for space-fractional diffusion equations , 2019, Numerical Algorithms.

[12]  Alvaro H. Salas,et al.  Exact solutions of coupled sine–Gordon equations , 2010 .

[13]  Hefeng Wu,et al.  A new geometric modeling approach for woven fabric based on Frenet frame and Spiral Equation , 2018, J. Comput. Appl. Math..

[14]  D. Liang,et al.  The energy-preserving finite difference methods and their analyses for system of nonlinear wave equations in two dimensions , 2020 .

[15]  K. Hosseini,et al.  New exact solutions of the coupled sine-Gordon equations in nonlinear optics using the modified Kudryashov method , 2018 .

[16]  Xiaofeng Yang,et al.  A novel linear second order unconditionally energy stable scheme for a hydrodynamic Q-tensor model of liquid crystals , 2017 .

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

[18]  Xiaofeng Yang,et al.  Numerical approximations for a three-component Cahn–Hilliard phase-field model based on the invariant energy quadratization method , 2017 .

[19]  Xiaoli Li,et al.  Efficient modified stabilized invariant energy quadratization approaches for phase-field crystal equation , 2019, Numerical Algorithms.

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

[21]  W. Strauss,et al.  Numerical solution of a nonlinear Klein-Gordon equation , 1978 .

[22]  G. Quispel,et al.  A new class of energy-preserving numerical integration methods , 2008 .

[23]  Wei Xiao,et al.  Global solutions and finite time blow up for some system of nonlinear wave equations , 2012, Appl. Math. Comput..

[24]  Brynjulf Owren,et al.  A General Framework for Deriving Integral Preserving Numerical Methods for PDEs , 2011, SIAM J. Sci. Comput..

[25]  Yuri S. Kivshar,et al.  Nonlinear dynamics of the Frenkel—Kontorova model , 1998 .

[26]  Zhiguo Xu,et al.  Error estimates in the energy space for a Gautschi-type integrator spectral discretization for the coupled nonlinear Klein-Gordon equations , 2016, J. Comput. Appl. Math..

[27]  M. Mirzazadeh,et al.  Exact solitons of the coupled sine-Gordon equation in nonlinear system , 2017 .

[28]  Jia Zhao,et al.  Numerical approximations for the molecular beam epitaxial growth model based on the invariant energy quadratization method , 2017, J. Comput. Phys..

[29]  S. Yomosa,et al.  Soliton excitations in deoxyribonucleic acid (DNA) double helices , 1983 .

[30]  Dingwen Deng.  Numerical Simulation of the Coupled Sine-Gordon Equations via a Linearized and Decoupled Compact ADI Method , 2019, Numerical Functional Analysis and Optimization.

[31]  Chaolong Jiang,et al.  A Linearly Implicit and Local Energy-Preserving Scheme for the Sine-Gordon Equation Based on the Invariant Energy Quadratization Approach , 2018, Journal of Scientific Computing.

[32]  Feng Liao,et al.  Two energy-conserving and compact finite difference schemes for two-dimensional Schrödinger-Boussinesq equations , 2020, Numerical Algorithms.

[33]  Zuntao Fu,et al.  The periodic solutions for a class of coupled nonlinear Klein–Gordon equations , 2004 .

[34]  Dmitry E. Pelinovsky,et al.  On the exchange of energy in coupled Klein-Gordon equations , 2003 .

[35]  Haochen Li,et al.  Partitioned averaged vector field methods , 2018, J. Comput. Phys..

[36]  Dong Liang,et al.  A new high-order energy-preserving scheme for the modified Korteweg-de Vries equation , 2016, Numerical Algorithms.