An energy-preserving Crank-Nicolson Galerkin method for Hamiltonian partial differential equations. (English) Zbl 1351.65098
Authors’ abstract: A semidiscretization based method for solving Hamiltonian partial differential equations (PDEs) is proposed. Our key idea consists of two approaches. First, the underlying equation is discretized in space via a selected finite element method and the Hamiltonian PDE can thus be casted to Hamiltonian ordinary differential equations (ODEs) based on the weak formulation of the system. Second, the resulting ordinary differential system is solved by an energy-preserving integrator. The relay leads to a fully discretized and energy-preserved scheme. This strategy is fully realized for solving a nonlinear Schrödinger equation through a combination of the Galerkin discretization in space and a Crank-Nicolson scheme in time. The order of convergence of our new method is \(\mathcal{O}(\tau^2+h^2)\) if the discrete \(L^{2}\)-norm is employed. An error estimate is acquired and analyzed without grid ratio restrictions. Numerical examples are given to further illustrate the conservation and convergence of the energy-preserving scheme constructed.
Reviewer: Francisco Pérez Acosta (La Laguna)
MSC:
| 65P10 | Numerical methods for Hamiltonian systems including symplectic integrators |
| 65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 37M15 | Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems |
Keywords:
energy-preserving schemes; error estimate; Hamiltonian equations; nonlinear Schrödinger equation; semidiscretization; Galerkin discretization in space; Crank-Nicolson scheme in timeReferences:
| [1] | K.Feng, Difference schemes for Hamiltonian formalism and symplectic geometry, J Comput Math4 (1986), 279-289. · Zbl 0596.65090 |
| [2] | A.Iserles, A first course in the numerical analysis of differential equations, 2nd Ed., Cambridge University Press, Cambridge, 2008. |
| [3] | K.Feng and M. Z.Qin, Symplectic Geometric Algorithms for Hamiltonian Systems, Springer‐Verlag/Zhejiang Science and Technology Publishing House, Berlin/Hangzhou, 2010. · Zbl 1207.65149 |
| [4] | E.Hairer, S. P.Nèrsett, and G.Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems, 2nd Ed., Springer‐Verlag, Berlin, 1993. · Zbl 0789.65048 |
| [5] | J. M.Sanz‐Serna and M. P.Calvo, Geometric Numerical Integration, Chapman & Hall, London (UK), 1994. · Zbl 0816.65042 |
| [6] | B.Leimkulher and S.Reich, Simulating Hamiltonian Dynamics, Cambridge University Press, Cambridge (UK), 2004. · Zbl 1069.65139 |
| [7] | L.Brugnano and F.Iavernaro, Line Integral Methods for Conservative Problems, Chapman & Hall/CRC, Boca Raton (FL), 2016. · Zbl 1335.65097 |
| [8] | C. W.Li and M. Z.Qin, A symplectic difference scheme for infinite dimensional Hamiltonian systems, J Comput Math6 (1988), 164-174. · Zbl 0669.70019 |
| [9] | R.McLachlan, Symplectic integration of Hamiltonian wave equations, Numer Math66 (1993), 465-492. · Zbl 0831.65099 |
| [10] | Q.Sheng, A.Khaliq, and E.Al‐Said, Solving the generalized nonlinear Schrödinger equation in quantum mechanics via quartic spline approximations, J Comput Phys166 (2001), 400-417. · Zbl 0979.65082 |
| [11] | T. J.Bridges and S.Reich, Multi‐symplectic spectral discretizations for the Zakharov‐Kuznetsov and shallow water equations, Physica D: Nonlinear Phenom152 (2001), 491-504. · Zbl 1032.76053 |
| [12] | J. B.Chen and M. Z.Qin, Multi‐symplectic fourier pseudospectral method for the nonlinear Schrödinger equation, Electron Trans Numer Anal12 (2001), 193-204. · Zbl 0980.65108 |
| [13] | B.Fornberg, A Practical Guide to Pseudospectral Methods, Cambridge University Press, 1996. · Zbl 0844.65084 |
| [14] | Y. S.Kivshar and G. P.Agrawal, Optical Solutions, Academic Press, San Diego, USA2003. |
| [15] | W.Bao and J.Shen, A fourth‐order time‐splitting Laguerre‐Hermite pseudo‐spectral method for Bose‐Einstein condensates, SIAM J Sci Comput26 (2005), 2010-2028. · Zbl 1084.35083 |
| [16] | M.Thalhammer, High‐order exponential operator splitting methods for timedependent Schrödinger equations, SIAM J Numer Anal46 (2008), 2022-2038. · Zbl 1170.65061 |
| [17] | Z.Sun and D.Zhao, On the L_∞ convergence of a difference scheme for coupled nonlinear Schrödinger equations, Comput Math Appl59 (2010), 3286-3300. · Zbl 1198.65173 |
| [18] | T.Wang, B.Guo, and Q.Xu, Fourth‐order compact and energy conservative difference schemes for the nonlinear Schrödinger equation in two dimensions, J Comput Phys243 (2013), 382-399. · Zbl 1349.65347 |
| [19] | LinghuaKong, JialinHong, LihaiJi, and PengfeiZhu, Compact and efficient conservative schemes for coupled nonlinear Schrödinger equations, Numer Methods Partial Differential Equations31 (2015), 1814-1843. · Zbl 1339.65125 |
| [20] | G.Akrivis, V.Dougalis, and O.Karakashian, On fully discrete Galerkin methods of second‐order temporal accuracy for the nonlinear Schrödinger equation, Numer Math59 (1991), 31-53. · Zbl 0739.65096 |
| [21] | O.Karakashian, G.Akrivis, and V.Dougalis, On optimal order error estimates for the nonlinear Schrödinger equation, SIAM J Numer Anal30 (1993), 377-400. · Zbl 0774.65091 |
| [22] | X.Liang, A.Q.M.Khaliq, and Y.Xing, Fourth order exponential time differencing method with local discontinuous Galerkin approximation for coupled nonlinear Schrödinger equations, Commun Comput Phys17 (2015), 510-514. · Zbl 1388.65086 |
| [23] | X.Liang, A. Q. M.Khaliq, and Q.Sheng, Exponential time differencing Crank‐Nicolson method with a quartic spline approximation for nonlinear Schrödinger equations, Appl Math Comput235 (2014), 235-252. · Zbl 1334.65134 |
| [24] | H.Bhatt and A. Q. M.Khaliq, Higher order exponential time differencing scheme for system of nonlinear Schrödinger equations, Appl Math Comput228 (2014), 271-291. · Zbl 1364.78033 |
| [25] | C.Besse, B.Bidegaray, and S.Descombes, Order estimates in time of splitting methods for the nonlinear Schrödinger equation, SIAM J Numer Anal40 (2002), 26-40. · Zbl 1026.65073 |
| [26] | O.Gonzalez, Time integration and discrete Hamiltonian systems, J Nonlinear Sci6 (1996), 449-467. · Zbl 0866.58030 |
| [27] | T.Matsuo, High‐order schemes for conservative or dissipative systems, J Comput Appl Math152 (2003), 305-317. · Zbl 1019.65042 |
| [28] | L.Brugnano, F.Iavernaro, and D.Trigiante, Hamiltonian boundary value methods (Energy preserving discrete line integral methods), J Numer Anal Ind Appl Math5 (2010), 17-37. · Zbl 1432.65182 |
| [29] | L.Brugnano, F.Iavernaro, and D.Trigiante, A simple framework for the derivation and analysis of effective one‐step methods for ODEs, Appl Math Comput218 (2012), 8475-485. · Zbl 1245.65086 |
| [30] | L.Brugnano, G. F.Caccia, and F.Iavernaro, Energy conservation issues in the numerical solution of the semilinear wave equation, Appl Math Comput270 (2015), 842-870. · Zbl 1410.65477 |
| [31] | G. R. W.Quispel and D. I.McLaren, A new class of energy‐preserving numerical integration methods, J Phys A: Math Theor41 (2008), 045206. · Zbl 1132.65065 |
| [32] | M. S.Ismail, Numerical solution of coupled nonlinear Schrödinger equation by Galerkin method, Math Comput Simulation78 (2008), 532-547. · Zbl 1145.65075 |
| [33] | Y. L.Zhou, Applications of Discrete Functional Analysis to the Finite Difference Method, International Academic Publishers, Beijing, 1990. |
| [34] | V.Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer, Berlin, 1984. · Zbl 0528.65052 |
| [35] | C.Canuto, M. Y.Hussaini, A.Quarteroni, M. Y.Hussaini, and T. A.Zang, Spectral Methods: Fundamentals in Single Domains, Springer, Berlin, 2006. · Zbl 1093.76002 |
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