Abstract
We consider a second-order SAV scheme for the nonlinear Schrödinger equation in the whole space with typical generalized nonlinearities, and carry out a rigorous error analysis. We also develop a fully discretized SAV scheme with Hermite–Galerkin approximation for the space variables, and present numerical experiments to validate our theoretical results.
Similar content being viewed by others
References
Abdullaev, F., Darmanyan, S., Khabibullaev, P., Engelbrecht, J.: Optical Solitons. Springer, Berlin (2014)
Akrivis, G., Dougalis, V., Karakashian, O.: On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear schrödinger equation. Numer. Math. 59(1), 31–53 (1991)
Antoine, X., Arnold, A., Besse, C., Ehrhardt, M., Schadle, A.: A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrodinger equations. Commun. Comput. Phys. 4(4), 729–796 (2008)
Antoine, X., Bao, W., Besse, C.: Computational methods for the dynamics of the nonlinear Schrödinger/Gross–Pitaevskii equations. Comput. Phys. Commun. 184(12), 2621–2633 (2013)
Antoine, X., Besse, C., Klein, P.: Absorbing boundary conditions for general nonlinear Schrödinger equations. SIAM J. Sci. Comput. 33(2), 1008–1033 (2011)
Antoine, X., Besse, C., Klein, P.: Numerical solution of time-dependent nonlinear Schrödinger equations using domain truncation techniques coupled with relaxation scheme. Laser Phys. 21(8), 1491–1502 (2011)
Antoine, X., Shen, J., Tang, Q.: Scalar auxiliary variable/lagrange multiplier based pseudospectral schemes for the dynamics of nonlinear Schrödinger/Gross–Pitaevskii equations (2020). https://hal.archives-ouvertes.fr/hal-02940080/document
Bao, W., Cai, Y.: Uniform error estimates of finite difference methods for the nonlinear Schrödinger equation with wave operator. SIAM J. Numer. Anal. 50(2), 492–521 (2012)
Bao, W., Shen, J.: A fourth-order time-splitting laguerre-hermite pseudospectral method for Bose–Einstein condensates. SIAM J. Sci. Comput. 26(6), 2010–2028 (2005)
Besse, C.: A relaxation scheme for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 42(3), 934–952 (2004)
Borzì, A., Decker, E.: Analysis of a leap-frog pseudospectral scheme for the Schrödinger equation. J. Comput. Appl. Math. 193(1), 65–88 (2006)
Cazenave, T.: Semilinear Schrödinger Equations, volume 10 of Courant Lecture Notes in Mathematics. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI (2003)
Cheng, Q., Shen, J.: Global constraints preserving scalar auxiliary variable schemes for gradient flows. SIAM J. Sci. Comput. 42(4), A2489–A2513 (2020)
Dehghan, M., Taleei, A.: Numerical solution of nonlinear Schrödinger equation by using time–space pseudo-spectral method. Numer. Methods Partial Differ. Equ. 26(4), 979–992 (2010)
Gardner, L.R.T., Gardner, G.A., Zaki, S.I., El-Sahrawi, Z.: B-spline finite element studies of the non-linear schrödinger equation. Comput. Methods Appl. Mech. Eng. 108(3–4), 303–318 (1993)
Guo, B., Shen, J., Xu, C.: Spectral and Pseudospectral Approximations Using Hermite Functions: Application to the Dirac Equation, vol. 19, pp. 35–55 (2003). Challenges in Computational Mathematics, Pohang (2001)
Ignat, L.I., Zuazua, E.: Numerical dispersive schemes for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 47(2), 1366–1390 (2009)
Lu, T., Cai, W.: Fourier spectral-discontinuous Galerkin method for time-dependent 3-d Schrodinger–Poisson equations with discontinuous potentials. J. Comput. Appl. Math. 220(1–2), 588–614 (2008)
Robinson, M.P.: The solution of nonlinear Schrödinger equations using orthogonal spline collocation. Comput. Math. Appl. 33(7), 39–57 (1997)
Schrödinger, E.: An undulatory theory of the mechanics of atoms and molecules. Phys. Rev. 28(6), 1049 (1926)
Shen, J., Xu, J.: Convergence and error analysis for the scalar auxiliary variable (SAV) schemes to gradient flows. SIAM J. Numer. Anal. 56(5), 2895–2912 (2018)
Shen, J., Xu, J., Yang, J.: The scalar auxiliary variable (SAV) approach for gradient fluids. J. Comput. Phys. 353, 407–416 (2018)
Shen, J., Xu, J., Yang, J.: A new class of efficient and robust energy stable schemes for gradient flows. SIAM Rev. 61(3), 474–506 (2019)
Shen, J., Tang, T., Wang, L.-L.: Spectral Methods: Algorithms, Analysis and Applications, vol. 41. Springer, Berlin (2011)
Sulem, C., Sulem, P.-L.: The Nonlinear Schrödinger Equation: Self-focusing and Wave Collapse, vol. 139. Springer, Berlin (2007)
Thalhammer, M.: High-order exponential operator splitting methods for time-dependent Schrödinger equations. SIAM J. Numer. Anal. 46(4), 2022–2038 (2008)
Wang, H.: An efficient Chebyshev–Tau spectral method for Ginzburg–Landau–Schrödinger equations. Comput. Phys. Commun. 181, 325–340 (2010)
Wang, J.: A new error analysis of Crank-Nicolson Galerkin fems for a generalized nonlinear Schrödinger equation. J. Sci. Comput. 60(2), 390–407 (2014)
Wang, J.: Unconditional stability and convergence of Crank–Nicolson Galerkin FEMs for a nonlinear Schrödinger–Helmholtz system. Numer. Math. 139(2), 479–503 (2018)
Zouraris, G.: On the convergence of a linear two-step finite element method for the nonlinear Schrodinger equation. ESAIM Math. Model. Numer. Anal. Model. Math. 35(3), 389–405 (2001)
Funding
The work of J. Shen is supported in part by NSF Grant DMS-2012585 and by AFOSR FA9550-20-1-0309. The work of Q. Zhuang is supported in part by National Natural Science Foundation of China (No. 11771083), and the Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of Huaqiao University(No. ZQN-702). Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Deng, B., Shen, J. & Zhuang, Q. Second-Order SAV Schemes for the Nonlinear Schrödinger Equation and Their Error Analysis. J Sci Comput 88, 69 (2021). https://doi.org/10.1007/s10915-021-01576-y
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10915-021-01576-y