Energy preserving integration of bi-Hamiltonian partial differential equations

2013-12-01
The energy preserving average vector field (AVF) integrator is applied to evolutionary partial differential equations (PDEs) in bi-Hamiltonian form with nonconstant Poisson structures. Numerical results for the Korteweg de Vries (KdV) equation and for the Ito type coupled KdV equation confirm the long term preservation of the Hamiltonians and Casimir integrals, which is essential in simulating waves and solitons. Dispersive properties of the AVF integrator are investigated for the linearized equations to examine the nonlinear dynamics after discretization.
APPLIED MATHEMATICS LETTERS

Suggestions

Energy preserving methods for lattice equations
Erdem, Özge; Karasözen, Bülent (2010-11-27)
Integral preserving methods, like the averaged vector field, discrete gradient and trapezoidal methods are to Poisson systems. Numerical experiments on the Volterra equations and integrable discretization of the nonlinear Schrodinger equation are presented.
HIGHER-DERIVATIVE EFFECTIVE YANG-MILLS THEORY AND STATIC SPHERICALLY SYMMETRICAL FIELD CONFIGURATIONS
BASKAL, S; DERELI, T (IOP Publishing, 1993-04-01)
The variational field equations and the covariantly conserved energy-momentum tensor of a higher-derivative effective Yang-Mills theory are given. A class of static spherically symmetric gauge field configurations that follow from the Wu-Yang ansatz is considered.
Quantum mechanical computation of billiard systems with arbitrary shapes
Erhan, İnci; Taşeli, Hasan; Department of Mathematics (2003)
An expansion method for the stationary Schrodinger equation of a particle moving freely in an arbitrary axisymmeric three dimensional region defined by an analytic function is introduced. The region is transformed into the unit ball by means of coordinate substitution. As a result the Schrodinger equation is considerably changed. The wavefunction is expanded into a series of spherical harmonics, thus, reducing the transformed partial differential equation to an infinite system of coupled ordinary differenti...
EXACT SPIN AND PSEUDO-SPIN SYMMETRIC SOLUTIONS OF THE DIRAC-KRATZER PROBLEM WITH A TENSOR POTENTIAL VIA LAPLACE TRANSFORM APPROACH
Arda, Altug; Sever, Ramazan (2012-09-28)
Exact bound state solutions of the Dirac equation for the Kratzer potential in the presence of a tensor potential are studied by using the Laplace transform approach for the cases of spin- and pseudo-spin symmetry. The energy spectrum is obtained in the closed form for the relativistic as well as non-relativistic cases including the Coulomb potential. It is seen that our analytical results are in agreement with the ones given in the literature. The numerical results are also given in a table for different p...
Average Vector Field Splitting Method for Nonlinear Schrodinger Equation
Akkoyunlu, Canan; Karasözen, Bülent (2012-05-02)
The energy preserving average vector field integrator is applied to one and two dimensional Schrodinger equations with symmetric split-step method. The numerical results confirm the long-term preservation of the Hamiltonians, which is essential in simulating periodic waves.
Citation Formats
B. Karasözen, “Energy preserving integration of bi-Hamiltonian partial differential equations,” APPLIED MATHEMATICS LETTERS, pp. 1125–1133, 2013, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/30940.