520 research outputs found
Poisson integrators
An overview of Hamiltonian systems with noncanonical Poisson structures is
given. Examples of bi-Hamiltonian ode's, pde's and lattice equations are
presented. Numerical integrators using generating functions, Hamiltonian
splitting, symplectic Runge-Kutta methods are discussed for Lie-Poisson systems
and Hamiltonian systems with a general Poisson structure. Nambu-Poisson systems
and the discrete gradient methods are also presented.Comment: 30 page
Linearly implicit exponential integrators for damped Hamiltonian PDEs
Structure-preserving linearly implicit exponential integrators are
constructed for Hamiltonian partial differential equations with linear constant
damping. Linearly implicit integrators are derived by polarizing the polynomial
terms of the Hamiltonian function and portioning out the nonlinearly of
consecutive time steps. They require only a solution of one linear system at
each time step. Therefore they are computationally more advantageous than
implicit integrators. We also construct an exponential version of the
well-known one-step Kahan's method by polarizing the quadratic vector field.
These integrators are applied to one-dimensional damped Burger's,
Korteweg-de-Vries, and nonlinear Schr\"odinger equations. Preservation of the
dissipation rate of linear and quadratic conformal invariants and the
Hamiltonian is illustrated by numerical experiments
Reduced-order modeling for Ablowitz-Ladik equation
In this paper, reduced-order models (ROMs) are constructed for the
Ablowitz-Ladik equation (ALE), an integrable semi-discretization of the
nonlinear Schr\"{o}dinger equation (NLSE) with and without damping. Both ALEs
are non-canonical conservative and dissipative Hamiltonian systems with the
Poisson matrix, depending quadratically on the state variables and with
quadratic Hamiltonian. The full-order solutions are obtained with the energy
preserving midpoint rule for the conservative ALE and exponential midpoint rule
for the dissipative ALE. The reduced-order solutions are constructed
intrusively by preserving the skew-symmetric structure of the reduced
non-canonical Hamiltonian system by applying proper orthogonal decomposition
with the Galerkin projection. For an efficient offline-online decomposition of
the ROMs, the quadratic nonlinear terms of the Poisson matrix are approximated
by the discrete empirical interpolation method. The computation of the
reduced-order solutions is further accelerated by the use of tensor techniques.
Preservation of the Hamiltonian and momentum for the conservative ALE, and
preservation of dissipation properties of the dissipative ALE, guarantee the
long-term stability of soliton solutions
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