An energy-preserving algorithm for nonlinear Hamiltonian wave equations with Neumann boundary conditions

In this paper, a novel energy-preserving numerical scheme for nonlinear Hamiltonian wave equations with Neumann boundary conditions is proposed and analyzed based on the blend of spatial discretization by finite element method (FEM) and time discretization by Average Vector Field (AVF) approach. We first use the finite element discretization in space, which leads to a system of Hamiltonian ODEs whose Hamiltonian can be thought of as the semi-discrete energy of the original continuous system. The stability of the semi-discrete finite element scheme is analyzed. We then apply the AVF approach to the Hamiltonian ODEs to yield a new and efficient fully discrete scheme, which can preserve exactly (machine precision) the semi-discrete energy. The blend of FEM and AVF approach derives a new and efficient numerical scheme for nonlinear Hamiltonian wave equations. The numerical results on a single-soliton problem and a sine-Gordon equation are presented to demonstrate the remarkable energy-preserving property of the proposed numerical scheme.

[1]  L. Vu-Quoc,et al.  Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein-Gordon equation , 1995 .

[2]  Bin Wang,et al.  Trigonometric collocation methods based on Lagrange basis polynomials for multi-frequency oscillatory second-order differential equations , 2016, J. Comput. Appl. Math..

[3]  Mehdi Dehghan,et al.  Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices , 2006, Math. Comput. Simul..

[4]  Antonella Zanna,et al.  Preserving algebraic invariants with Runge-Kutta methods , 2000 .

[5]  Bin Wang,et al.  Arbitrary-Order Trigonometric Fourier Collocation Methods for Multi-Frequency Oscillatory Systems , 2016, Found. Comput. Math..

[6]  Bin Wang,et al.  Sixth-order symplectic and symmetric explicit ERKN schemes for solving multi-frequency oscillatory nonlinear Hamiltonian equations , 2017 .

[7]  J. Lambert,et al.  Symmetric Multistip Methods for Periodic Initial Value Problems , 1976 .

[8]  Patrick Joly,et al.  Energy preserving schemes for nonlinear Hamiltonian systems of wave equations: Application to the vibrating piano string , 2010 .

[9]  J. Butcher.  The Numerical Analysis of Ordinary Di erential Equa-tions , 1986 .

[10]  Abdul-Majid Wazwaz,et al.  New travelling wave solutions to the Boussinesq and the Klein–Gordon equations , 2008 .

[11]  J. Gibbon,et al.  Solitons and Nonlinear Wave Equations , 1982 .

[12]  G. Quispel,et al.  Geometric integration using discrete gradients , 1999, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[13]  G. R. W. Quispel,et al.  Linearization-preserving self-adjoint and symplectic integrators , 2009 .

[14]  Elena Celledoni,et al.  Preserving energy resp. dissipation in numerical PDEs using the "Average Vector Field" method , 2012, J. Comput. Phys..

[15]  Mehdi Dehghan,et al.  Numerical solution of the Klein–Gordon equation via He’s variational iteration method , 2007 .

[16]  Mehdi Dehghan,et al.  High-order solution of one-dimensional sine-Gordon equation using compact finite difference and DIRKN methods , 2010, Math. Comput. Model..

[17]  J. Marsden,et al.  Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators , 1988 .

[18]  Arieh Iserles,et al.  B-SERIES METHODS CANNOT BE VOLUME-PRESERVING , 2007 .

[19]  L. Shampine.  Conservation laws and the numerical solution of ODEs, II , 1999 .

[20]  Ernst Hairer,et al.  ON ENERGY CONSERVATION OF THE SIMPLIFIED TAKAHASHI-IMADA METHOD , 2009 .

[21]  G. Quispel,et al.  A new class of energy-preserving numerical integration methods , 2008 .

[22]  Mehdi Dehghan,et al.  Fourth-order compact solution of the nonlinear Klein-Gordon equation , 2009, Numerical Algorithms.

[23]  Mehdi Dehghan,et al.  Numerical solution to the unsteady two‐dimensional Schrödinger equation using meshless local boundary integral equation method , 2008 .

[24]  A. G. Bratsos.  On the numerical solution of the Klein-Gordon equation , 2009 .

[25]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

[26]  Xinyuan Wu,et al.  An extended discrete gradient formula for oscillatory Hamiltonian systems , 2013 .

[27]  Masami Okada,et al.  Numerical simulation for a nonlinear partial differential equation with variable coefficients by means of the discrete variational derivative method , 2006 .

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

[29]  Anjan Biswas,et al.  Soliton perturbation theory for phi-four model and nonlinear Klein–Gordon equations , 2009 .

[30]  Bin Wang,et al.  Efficient energy-preserving integrators for oscillatory Hamiltonian systems , 2013, J. Comput. Phys..