nach oben

Journal of Scientific Computing

Erschienen in:

17.04.2018

On the Conservation of Fractional Nonlinear Schrödinger Equation’s Invariants by the Local Discontinuous Galerkin Method

verfasst von: P. Castillo, S. Gómez

Erschienen in: Journal of Scientific Computing | Ausgabe 3/2018

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

Using the primal formulation of the Local Discontinuous Galerkin (LDG) method, discrete analogues of the energy and the Hamiltonian of a general class of fractional nonlinear Schrödinger equation are shown to be conserved for two stabilized version of the method. Accuracy of these invariants is numerically studied with respect to the stabilization parameter and two different projection operators applied to the initial conditions. The fully discrete problem is analyzed for two implicit time step schemes: the midpoint and the modified Crank–Nicolson; and the explicit circularly exact Leapfrog scheme. Stability conditions for the Leapfrog scheme and a stabilized version of the LDG method applied to the fractional linear Schrödinger equation are derived using a von Neumann stability analysis. A series of numerical experiments with different nonlinear potentials are presented.

Vorheriger Artikel Hybrid Discretization Methods with Adaptive Yield Surface Detection for Bingham Pipe Flows

Nächster Artikel Enriched Spectral Methods and Applications to Problems with Weakly Singular Solutions

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Aboelenen, T.: A high-order nodal discontinuous Galerkin method for nonlinear fractional Schrödinger type equations. Commun. Nonlinear Sci. Numer. Simul. 54, 428–452 (2018)MathSciNetCrossRef

Ardila, A.: Existence and stability of standing waves for nonlinear fractional Schrödinger equation with logarithmic nonlinearity. Nonlinear Anal. 155, 52–64 (2017)MathSciNetCrossRef

Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19, 742–760 (1982)MathSciNetCrossRef

Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749–1779 (2002)MathSciNetCrossRef

Bhrawy, A.H., Zaky, M.A.: Highly accurate numerical schemes for multi-dimensional space variable-order fractional Schrödinger equations. Computers & Mathematics with Applications 73(6), 1100–1117 (2017). (Advances in Fractional Differential Equations (IV): Time-fractional PDEs) MathSciNetCrossRef

Bhrawy, A.H., Zaky, M.A.: An improved collocation method for multi-dimensional spacetime variable-order fractional Schrödinger equations. Appl. Numer. Math. 111, 197–218 (2017)MathSciNetCrossRef

Castillo, P.: An optimal error estimate for the local discontinuous Galerkin method. In: Cockburn, B., Karniadakis, G.E., Shu, C.-W. (eds.) Discontinuous Galerkin Methods: Theory, Computation and Applications, Volume 11 of Lecture Notes in Computational Science and Engineering, pp. 285–290. Springer, Berlin (2000)CrossRef

Castillo, P., Cockburn, B., Perugia, I., Schötzau, D.: An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal. 38(5), 1676–1706 (2000)MathSciNetCrossRef

Castillo, P., Cockburn, B., Schötzau, D., Schwab, Ch.: An optimal a priori error estimate for the \(hp\)-version of the local discontinuous Galerkin method for convection-diffusion problems. Math. Comput. 71(238), 455–478 (2001)MathSciNetCrossRef

10.

Cheng, Y., Shu, C.W.: A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives. Math. Comput. 77(262), 699–730 (2008)MathSciNetCrossRef

11.

Cockburn, B., Shu, C.W.: The local discontinuous Galerkin method for time-dependent convection–diffusion systems. SIAM J. Numer. Anal. 35(6), 2440–2463 (1998)MathSciNetCrossRef

12.

D’Avenia, P., Squassina, M., Zenari, M.: Fractional logarithmic Schrödinger equations. Math. Methods Appl. Sci. 38(18), 5207–5216 (2015)MathSciNetCrossRef

13.

Delfour, M., Fortin, M., Payré, G.: Finite-difference solutions of a non-linear Schrödinger equation. J. Comput. Phys. 44(2), 277–288 (1981)MathSciNetCrossRef

14.

Deng, W.H.: Finite element method for the space and time fractional Fokker–Planck equation. SIAM J. Numer. Anal. 47(1), 204–226 (2008)MathSciNetCrossRef

15.

Deng, W.H., Hesthaven, J.S.: Local Discontinuous Galerkin methods for fractional diffusion equations. ESAIM: M2AN 47(6), 1845–1864 (2013)MathSciNetCrossRef

16.

Griffiths, D.F., Mitchell, A.R., Morris, JLi: A numerical study of the nonlinear Schrödinger equation. Comput. Methods Appl. Mech. Eng. 45(1), 177–215 (1984)CrossRef

17.

Guo, B., Han, Y., Xin, J.: Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation. Appl. Math. Comput. 204(1), 468–477 (2008)MathSciNetMATH

18.

Guo, X., Xu, M.: Some physical applications of fractional Schrödinger equation. J. Math. Phys. 47(8), 082104 (2006)MathSciNetCrossRef

19.

Herbst, B.M., Morris, JLi, Mitchell, A.R.: Numerical experience with the nonlinear Schrödinger equation. J. Comput. Phys. 60, 282–305 (1985)MathSciNetCrossRef

20.

Klein, C., Sparber, C., Markowich, P.: Numerical study of fractional nonlinear Schrödinger equations. Proc. R. Soc. A 470, 20140364 (2014)MathSciNetCrossRef

21.

Laskin, N.: Fractional quantum mechanics. Phys. Rev. E 62(3), 3135 (2000)MathSciNetCrossRef

22.

Li, M., Gu, X.M., Huang, C., Fei, M., Zhang, G.: A fast linearized conservative finite element method for the strongly coupled nonlinear fractional equations. J. Comput. Phys. 358(1), 256–282 (2018)MathSciNetCrossRef

23.

Li, M., Huang, C., Wang, P.: Galerkin finite element method for nonlinear fractional Schrödinger equations. Numer. Algorithms 74(2), 499–525 (2017)MathSciNetCrossRef

24.

Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198. Academic press, Cambridge (1998)MATH

25.

Ran, M., Zhang, C.: A conservative difference scheme for solving the strongly coupled nonlinear fractional Schrödinger equations. Commun. Nonlinear Sci. Numer. Simul. 41, 64–83 (2016)MathSciNetCrossRef

26.

Sanz-Serna, J.M.: Methods for the numerical solution of the nonlinear Schrödinger equation. Math. Comput. 43(167), 21–27 (1984)CrossRef

27.

Sanz-Serna, J.M., Manoranjan, V.S.: A method for the integration in time of certain partial differential equations. J. Comput. Phys. 52(2), 273–289 (1983)MathSciNetCrossRef

28.

Sanz-Serna, J.M., Verwer, J.G.: Conservative and nonconservative schemes for the solution of the nonlinear Schrödinger equation. IMA J. Numer. Anal. 6, 25–42 (1986)MathSciNetCrossRef

29.

Shabat, A., Zakharov, V.: Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP 34(1), 62 (1972)MathSciNet

30.

Strauss, W., Vazquez, L.: Numerical solution of a nonlinear Klein–Gordon equation. J. Comput. Phys. 28(2), 271–278 (1978)MathSciNetCrossRef

31.

Sulem, C., Sulem, P.L.: The Nonlinear Schrödinger Equation, Volume 139 of Applied Mathematical Sciences. Springer, Berlin (1999)MATH

32.

Verwer, J.G., Dekker, K.: Step by step stability in the numerical solution of partial differential equations. Technical Report 161-83, Centre for Mathematics and Computer Science, Amsterdam (1983)

33.

Wang, D., Xiao, A., Yang, W.: Crank–Nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative. J. Comput. Phys. 242(1), 670–681 (2013)MathSciNetCrossRef

34.

Wang, P., Huang, C.: An energy conservative difference scheme for the nonlinear fractional Schrödinger equations. J. Comput. Phys. 293, 238–251 (2015)MathSciNetCrossRef

35.

Wang, P., Huang, C.: Split-step alternating direction implicit difference scheme for the fractional Schrödinger equation in two dimensions. Comput. Math. Appl. 71(5), 1114–1128 (2016)MathSciNetCrossRef

36.

Wei, L., Zhang, X., Kumar, S., Yildirim, A.: A numerical study based on an implicit fully discrete local discontinuous galerkin method for the time-fractional coupled Schrödinger system. Comput. Math. Appl. 64(8), 2603–2615 (2012)MathSciNetCrossRef

37.

Weideman, J.A.C., Herbst, B.M.: Split-step alternating direction implicit difference scheme for the fractional schrödinger equation in two dimensions. SIAM J. Numer. Anal. 23(3), 485–507 (1986)MathSciNetCrossRef

38.

Xu, Q., Hesthaven, J.S.: Discontinuous Galerkin method for fractional convection-diffusion equations. SIAM J. Numer. Anal. 52(1), 405–423 (2014)MathSciNetCrossRef

39.

Xu, Y., Shu, C.W.: Local discontinuous Galerkin methods for nonlinear Schrödinger equations. J. Comput. Phys. 205(1), 72–97 (2005)MathSciNetCrossRef

40.

Xu, Y., Shu, C.W.: Optimal error estimates of the semidiscrete local discontinuous Galerkin methods for high order wave equations. SIAM J. Numer. Anal. 50(1), 79–104 (2012)MathSciNetCrossRef

41.

Zhang, H., Hu, Q.: Existence of the global solution for fractional logarithmic Schrödinger equation. Comput. Math. Appl. 75(1), 161–169 (2018)MathSciNetCrossRef

Titel: On the Conservation of Fractional Nonlinear Schrödinger Equation’s Invariants by the Local Discontinuous Galerkin Method
verfasst von: P. Castillo
S. Gómez
Publikationsdatum: 17.04.2018
Verlag: Springer US
Erschienen in: Journal of Scientific Computing / Ausgabe 3/2018
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-018-0708-8

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Weitere Artikel der Ausgabe 3/2018

A Finite Element Method with Strong Mass Conservation for Biot’s Linear Consolidation Model

Fast Numerical Integration on Polytopic Meshes with Applications to Discontinuous Galerkin Finite Element Methods

A Strongly Conservative Hybrid DG/Mixed FEM for the Coupling of Stokes and Darcy Flow

Upscaled HDG Methods for Brinkman Equations with High-Contrast Heterogeneous Coefficient

Discontinuous Galerkin Methods for Acoustic Wave Propagation in Polygons

Non-conforming Harmonic Virtual Element Method: - and -Versions

Premium Partner