Explicit multi-symplectic extended leap-frog methods for Hamiltonian wave equations

In this paper, we study the integration of Hamiltonian wave equations whose solutions have oscillatory behaviors in time and/or space. We are mainly concerned with the research for multi-symplectic extended Runge-Kutta-Nystrom (ERKN) discretizations and the corresponding discrete conservation laws. We first show that the discretizations to the Hamiltonian wave equations using two symplectic ERKN methods in space and time respectively lead to an explicit multi-symplectic integrator (Eleap-frogI). Then we derive another multi-symplectic discretization using a symplectic ERKN method in time and a symplectic partitioned Runge-Kutta method, which is equivalent to the well-known Stormer-Verlet method in space (Eleap-frogII). These two new multi-symplectic schemes are extensions of the leap-frog method. The numerical stability and dispersive properties of the new schemes are analyzed. Numerical experiments with comparisons are presented, where the two new explicit multi-symplectic methods and the leap-frog method are applied to the linear wave equation and the Sine-Gordon equation. The numerical results confirm the superior performance and some significant advantages of our new integrators in the sense of structure preservation.

[1]  S. Reich,et al.  Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity , 2001 .

[2]  C. M. Schober,et al.  Dispersive properties of multisymplectic integrators , 2008, J. Comput. Phys..

[3]  Chun Li,et al.  Multi-symplectic Runge-Kutta methods for nonlinear Dirac equations , 2006 .

[4]  R. McLachlan.  Symplectic integration of Hamiltonian wave equations , 1993 .

[5]  Theodore E. Simos,et al.  New modified Runge-Kutta-Nyström methods for the numerical integration of the Schrödinger equation , 2010, Comput. Math. Appl..

[6]  T. Bridges.  Multi-symplectic structures and wave propagation , 1997, Mathematical Proceedings of the Cambridge Philosophical Society.

[7]  T. E. Simos,et al.  Closed Newton–Cotes trigonometrically-fitted formulae of high order for the numerical integration of the Schrödinger equation , 2008 .

[8]  K. Morton,et al.  Numerical Solution of Partial Differential Equations: Introduction , 2005 .

[9]  Jianlin Xia,et al.  Explicit symplectic multidimensional exponential fitting modified Runge-Kutta-Nyström methods , 2012 .

[10]  E. Hairer,et al.  Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations , 2004 .

[11]  A. Iserles.  A First Course in the Numerical Analysis of Differential Equations: Stiff equations , 2008 .

[12]  T. Monovasilis,et al.  New Second-order Exponentially and Trigonometrically Fitted Symplectic Integrators for the Numerical Solution of the Time-independent Schrödinger Equation , 2007 .

[13]  Xinyuan Wu,et al.  Extended RKN-type methods for numerical integration of perturbed oscillators , 2009, Comput. Phys. Commun..

[14]  Jianlin Xia,et al.  Order conditions for ARKN methods solving oscillatory systems , 2009, Comput. Phys. Commun..

[15]  U. Ascher,et al.  Multisymplectic box schemes and the Korteweg{de Vries equation , 2004 .

[16]  Theodore E. Simos.  High-order closed Newton-Cotes trigonometrically-fitted formulae for long-time integration of orbital problems , 2008, Comput. Phys. Commun..

[17]  L. Trefethen.  Group velocity in finite difference schemes , 1981 .

[18]  Xinyuan Wu,et al.  A trigonometrically fitted explicit Numerov-type method for second-order initial value problems with oscillating solutions , 2008 .

[19]  Pablo Martín,et al.  A new family of Runge–Kutta type methods for the numerical integration of perturbed oscillators , 1999, Numerische Mathematik.

[20]  Yongzhong Song,et al.  On multi-symplectic partitioned Runge-Kutta methods for Hamiltonian wave equations , 2006, Appl. Math. Comput..

[21]  Wei Shi,et al.  On symplectic and symmetric ARKN methods , 2012, Comput. Phys. Commun..

[22]  Matthias J. Ehrhardt,et al.  Geometric Numerical Integration Structure-Preserving Algorithms for QCD Simulations , 2012 .

[23]  T. E. Simos.  Closed Newton-Cotes Trigonometrically-Fitted Formulae for the Solution of the Schrodinger Equation , 2008 .

[24]  K. Morton,et al.  Numerical Solution of Partial Differential Equations , 1995 .

[25]  Hongyu Liu,et al.  Multi-symplectic Runge–Kutta-type methods for Hamiltonian wave equations , 2006 .

[26]  Jason Frank,et al.  Linear PDEs and Numerical Methods That Preserve a Multisymplectic Conservation Law , 2006, SIAM J. Sci. Comput..

[27]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[28]  Xinyuan Wu,et al.  Note on derivation of order conditions for ARKN methods for perturbed oscillators , 2009, Comput. Phys. Commun..

[29]  Xinyuan Wu,et al.  Trigonometrically fitted explicit Numerov-type method for periodic IVPs with two frequencies , 2008, Comput. Phys. Commun..

[30]  Bin Wang,et al.  ERKN methods for long-term integration of multidimensional orbital problems , 2013 .

[31]  Jialin Hong.  A Survey of Multi-symplectic Runge-Kutta Type Methods for Hamiltonian Partial Differential Equations , 2006 .

[32]  J. Marsden,et al.  Multisymplectic Geometry, Variational Integrators, and Nonlinear PDEs , 1998, math/9807080.

[33]  Zacharoula Kalogiratou,et al.  Computation of the eigenvalues of the Schrödinger equation by symplectic and trigonometrically fitted symplectic partitioned Runge Kutta methods , 2008 .

[34]  Robert I. McLachlan,et al.  On Multisymplecticity of Partitioned Runge-Kutta Methods , 2008, SIAM J. Sci. Comput..

[35]  Xinyuan Wu,et al.  Trigonometrically-fitted ARKN methods for perturbed oscillators , 2008 .

[36]  Bin Wang,et al.  ERKN integrators for systems of oscillatory second-order differential equations , 2010, Comput. Phys. Commun..

[37]  T. Bridges.  A geometric formulation of the conservation of wave action and its implications for signature and the classification of instabilities , 1997, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[38]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[39]  Xinyuan Wu,et al.  A note on stability of multidimensional adapted Runge–Kutta–Nyström methods for oscillatory systems☆ , 2012 .

[40]  Geng Sun,et al.  The multi-symplecticity of partitioned Runge-Kutta methods for Hamiltonian PDEs , 2005, Math. Comput..

[41]  Theodore E. Simos,et al.  Closed Newton-Cotes trigonometrically-fitted formulae of high order for long-time integration of orbital problems , 2009, Appl. Math. Lett..

[42]  S. Reich.  Multi-Symplectic Runge—Kutta Collocation Methods for Hamiltonian Wave Equations , 2000 .

[43]  E. Hairer,et al.  Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems , 1993 .

[44]  J. M. Franco.  Runge–Kutta–Nyström methods adapted to the numerical integration of perturbed oscillators , 2002 .

[45]  Ernst Hairer,et al.  Long-Time Energy Conservation of Numerical Methods for Oscillatory Differential Equations , 2000, SIAM J. Numer. Anal..