Explicit high-order structure-preserving algorithms for the two-dimensional fractional nonlinear Schrödinger equation

References

• X. Antoine, W. Bao, and C. Besse, Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations, Comput. Phys. Commun. 184 (2013), pp. 2621–2633.
   Web of Science ®Google Scholar
• J. Cai and J. Shen, Two classes of linearly implicit local energy-preserving approach for general multi-symplectic Hamiltonian PDEs, J. Comput. Phys. 401 (2020), p. 108975.
   Google Scholar
• J. Cai and H. Zhang, Efficient schemes for the damped nonlinear Schrödinger equation in high dimensions, Appl. Math. Lett. 102 (2020), p. 106158.
   Google Scholar
• W. Cai, H. Li, and Y. Wang, Partitioned averaged vector field methods, J. Comput. Phys. 370 (2018), pp. 25–42.
   Web of Science ®Google Scholar
• W. Cai, C. Jiang, Y. Wang, and Y. Song, Structure-preserving algorithms for the two-dimensional sine-Gordon equation with Neumann boundary conditions, J. Comput. Phys. 395 (2019), pp. 166–185.
   Web of Science ®Google Scholar
• M. Calvo, D. Hernndez-Abreu, J.I. Montijano, and L. Rndez, On the preservation of invariants by explicit Runge-Kutta methods, SIAM J. Sci. Comput. 28 (2006), pp. 868–885.
   Web of Science ®Google Scholar
• P. Constantin and J. Wu, An extension problem related to the fractional Laplacian, Commun. Partial Differ. Equ. 26 (2009), pp. 159–180.
   Google Scholar
• N. Del Buono and C. Mastroserio, Explicit methods based on a class of four stage fourth order Runge-Kutta methods for preserving quadratic laws, J. Comput. Appl. Math. 140 (2002), pp. 231–243.
   Web of Science ®Google Scholar
• S. Duo and Y. Zhang, Mass-conservative Fourier spectral methods for solving the fractional nonlinear Schrödinger equation, Comput. Math. Appl. 71 (2016), pp. 2257–2271.
   Web of Science ®Google Scholar
• Y. Fu, Y. Song, and Y. Wang, Maximum-norm error analysis of a conservative scheme for the damped nonlinear fractional Schrödinger equation, Math. Comput. Simul. 166 (2019), pp. 206–223.
   Web of Science ®Google Scholar
• Y. Fu, W. Cai, and Y. Wang, An explicit structure-preserving algorithm for the nonlinear fractional Hamiltonian wave equation, Appl. Math. Lett. 102 (2020), p. 106123.
   Google Scholar
• Y. Gong, J. Zhao, X. Yang, and Q. Wang, Fully discrete second-order linear schemes for hydrodynamic phase field models of binary viscous fluid flows with variable densities, SIAM J. Sci. Comput. 40 (2018), pp. B138–B167.
   Web of Science ®Google Scholar
• X. Gu and S. Wu, A parallel-in-time iterative algorithm for Volterra partial integral-differential problems with weakly singular kernel, J. Comput. Phys. 417 (2020), p. 109576.
   Google Scholar
• B. Guo and Z. Huo, Global well-posedness for the fractional nonlinear Schrödinger equation, Comm. Partial Differ. Equ. 36 (2010), pp. 247–255.
   Web of Science ®Google Scholar
• X. Guo and M. Xu, Some physical applications of fractional Schrödinger equation, J. Math. Phys.47 (2006), p. 082104.
   Web of Science ®Google Scholar
• B. Guo, Y. Han, and J. Xin, Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation, Appl. Math. Comput. 204 (2008), pp. 468–477.
   Web of Science ®Google Scholar
• E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration: Structure-preserving Algorithms for Ordinary Differential Equations, 2nd ed., Springer-Verlag, Berlin, 2006.
   Google Scholar
• D. Hu, W. Cai, Y. Song, and Y. Wang, A fourth-order dissipation-preserving algorithm with fast implementation for space fractional nonlinear damped wave equations, Commun. Nonlinear Sci. Numer. Simul. 91 (2020), p. 105432.
   Google Scholar
• D. Hu, W. Cai, Y. Fu, and Y. Wang, Fast dissipation-preserving difference scheme for nonlinear generalized wave equations with the integral fractional Laplacian, Commun. Nonlinear Sci. Numer. Simul. 99 (2021), p. 105786.
   Google Scholar
• Y. Huang, X. Li, and A. Xiao, Fourier pseudo spectral method on generalized sparse grids for the space-fractional Schrödinger equation, Comput. Math. Appl. 75 (2018), pp. 4241–4255.
   Web of Science ®Google Scholar
• C. Jiang, W. Cai, and Y. Wang, A linearly implicit and local energy-preserving scheme for the sine-Gordon equation based on the invariant energy quadratization approach, J. Sci. Comput. 80 (2019), pp. 1629–1655.
   Web of Science ®Google Scholar
• K. Kirkpatrick, E. Lenzmann, and G. Staffilani, On the continuum limit for discrete NLS with long-range lattice interactions, Commun. Math. Phys. 317 (2013), pp. 563–591.
   Web of Science ®Google Scholar
• N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A. 268 (2000), pp. 298–305.
   Web of Science ®Google Scholar
• N. Laskin, Fractional Schrödinger equation, Phys. Rev. 66 (2002), pp. 249–264.
   Google Scholar
• M. Li, X. Gu, C. Huang, and M. Fei, A fast linearized conservative finite element method for the strongly coupled nonlinear fractional Schrödinger equations, J. Comput. Phys. 358 (2018), pp. 256–282.
   Web of Science ®Google Scholar
• S. Longhi, Fractional Schrödinger equation in optics, Opt. Lett. 40 (2015), pp. 1117–1120.
   PubMed Web of Science ®Google Scholar
• J. Macías-Díaz, An explicit dissipation-preserving method for Riesz space-fractional nonlinear wave equations in multiple dimensions, Comput. Math. Appl. 59 (2018), pp. 67–87.
   Google Scholar
• M. Ran and C. Zhang, A conservative difference scheme for solving the strongly coupled nonlinear fractional Schrödinger equations, Commun. Nonlinear Sci. Numer. Simul. 41 (2016), pp. 64–83.
   Web of Science ®Google Scholar
• L. Roncal and P. Stinga, Fractional Laplacian on the torus, Commun. Contemp. Math. 18 (2016), p. 1550033.
   Google Scholar
• P Wang and C Huang, An energy conservative difference scheme for the nonlinear fractional Schrödinger equations, J. Comput. Phys. 293 (2015), pp. 238–251.
   Web of Science ®Google Scholar
• P. Wang and C. Huang, Structure-preserving numerical methods for the fractional Schrödinger equation, Appl. Numer. Math. 129 (2018), pp. 137–158.
   Web of Science ®Google Scholar
• D. Wang, A. Xiao, and W. Yang, A linearly implicit conservative difference scheme for the space fractional coupled nonlinear Schrödinger equations, J. Comput. Phys. 272 (2014), pp. 644–655.
   Web of Science ®Google Scholar
• A. Xiao and J. Wang, Symplectic scheme for the Schrödinger equation with fractional Laplacian, Appl. Numer. Math. 146 (2019), pp. 469–487.
   Web of Science ®Google Scholar
• X. Yang, J. Zhao, and Q. Wang, Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends, J. Comput. Phys. 327 (2016), pp. 294–316.
   Web of Science ®Google Scholar
• X. Yang, J. Zhao, and Q. Wang, Numerical approximations for the molecular beam epitaxial growth model based on the invariant energy quadratization method, J. Comput. Phys. 333 (2017), pp. 104–127.
   Web of Science ®Google Scholar
• W. Zeng, A. Xiao, and X. Li, Error estimate of Fourier pseudo-spectral method for multidimensional nonlinear complex fractional Ginzburg-Landau equations, Appl. Math. Lett. 93 (2019), pp. 40–45.
   Web of Science ®Google Scholar
• Y. Zhang, X. Liu, M.R. Belić, W. Zhong, Y. Zhang, and M. Xiao, Propagation dynamics of a light beam in a fractional nonlinear Schrödinger equation, Phys. Rev. Lett. 115 (2015), p. 180403.
   Google Scholar
• H. Zhang, X. Qian, J. Yan, and S. Song, Highly efficient invariant-conserving explicit Runge-Kutta schemes for nonlinear Hamiltonian differential equations, J. Comput. Phys. 418 (2020), p. 109598.
   Google Scholar
• X. Zhao, Z. Sun, and Z. Hao, A fourth-order compact ADI scheme for 2D nonlinear space fractional Schrödinger equation, SIAM J. Sci. Comput. 36 (2014), pp. A2865–A2886.
   Web of Science ®Google Scholar
• J. Zhao, Q Wang, and X Yang, Numerical approximations for a phase field dendritic crystal growth model based on the invariant energy quadratization approach, Int. J. Numer. Methods Eng. 110 (2017), pp. 279–300.
   Web of Science ®Google Scholar
• J. Zhao, Y. Li, and Y. Xu, An explicit fourth-order energy-preserving scheme for Riesz space fractional nonlinear wave equations, Appl. Math. Comput. 351 (2019), pp. 124–138.
   Web of Science ®Google Scholar