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Register a new userOn the symplectic integration of the discrete nonlinear Schrodinger equation with disorder
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We present several methods, which utilize symplectic integration techniques based on two and three part operator splitting, for numerically solving the equations of motion of the disordered, discrete nonlinear Schrodinger (DDNLS) equation, and compare their efficiency. Our results suggest that the most suitable methods for the very long time integration of this one-dimensional Hamiltonian lattice model with many degrees of freedom (of the order of a few hundreds) are the ones based on three part splits of the systems Hamiltonian. Two part split techniques can be preferred for relatively small lattices having up to $Napprox;$70 sites. An advantage of the latter methods is the better conservation of the systems second integral, i.e. the wave packets norm.
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We implement several symplectic integrators, which are based on two part splitting, for studying the chaotic behavior of one- and two-dimensional disordered Klein-Gordon lattices with many degrees of freedom and investigate their numerical performance. For this purpose, we perform extensive numerical simulations by considering many different initial energy excitations and following the evolution of the created wave packets in the various dynamical regimes exhibited by these models. We compare the efficiency of the considered integrators by checking their ability to correctly reproduce several features of the wave packets propagation, like the characteristics of the created energy distribution and the time evolution of the systems maximum Lyapunov exponent estimator. Among the tested integrators the fourth order $ABA864$ scheme cite{BCFLMM13} showed the best performance as it needed the least CPU time for capturing the correct dynamical behavior of all considered cases when a moderate accuracy in conserving the systems total energy value was required. Among the higher order schemes used to achieve a better accuracy in the energy conservation, the sixth order scheme $s11ABA82_6$ exhibited the best performance.
With Monte Carlo methods, we investigate the universality class of the depinning transition in the two-dimensional Ising model with quenched random fields. Based on the short-time dynamic approach, we accurately determine the depinning transition field and both static and dynamic critical exponents. The critical exponents vary significantly with the form and strength of the random fields, but exhibit independence on the updating schemes of the Monte Carlo algorithm. From the roughness exponents $zeta, zeta_{loc}$ and $zeta_s$, one may judge that the depinning transition of the random-field Ising model belongs to the new dynamic universality class with $zeta eq zeta_{loc} eq zeta_s$ and $zeta_{loc} eq 1$. The crossover from the second-order phase transition to the first-order one is observed for the uniform distribution of the random fields, but it is not present for the Gaussian distribution.
We provide a systematic comparison of two numerical methods to solve the widely used nonlinear Schrodinger equation. The first one is the standard second order split-step (SS2) method based on operator splitting approach. The second one is the Hamiltonian integration method (HIM). It allows the exact conservation of the Hamiltonian at the cost of requiring the implicit time stepping. We found that numerical error for HIM method is systematically smaller than the SS2 solution for the same time step. At the same time, one can take orders of magnitude larger time steps in HIM compared with SS2 still ensuring numerical stability. In contrast, SS2 time step is limited by the numerical stability threshold.
The time-dependent radiation transport equation is discretized using the meshless-local Petrov-Galerkin method with reproducing kernels. The integration is performed using a Voronoi tessellation, which creates a partition of unity that only depends on the position and extent of the kernels. The resolution of the integration automatically follows the particles and requires no manual adjustment. The discretization includes streamline-upwind Petrov-Galerkin stabilization to prevent oscillations and improve numerical conditioning. The angular quadrature is selectively refineable to increase angular resolution in chosen directions. The time discretization is done using backward Euler. The transport solve for each direction and the solve for the scattering source are both done using Krylov iterative methods. Results indicate first-order convergence in time and second-order convergence in space for linear reproducing kernels.
We present a novel framework for performing statistical sampling, expectation estimation, and partition function approximation using emph{arbitrary} heuristic stochastic processes defined over discrete state spaces. Using a highly parallel construction we call the emph{sequential constraining process}, we are able to simultaneously generate states with the heuristic process and accurately estimate their probabilities, even when they are far too small to be realistically inferred by direct counting. After showing that both theoretically correct importance sampling and Markov chain Monte Carlo are possible using the sequential constraining process, we integrate it into a methodology called emph{state space sampling}, extending the ideas of state space search from computer science to the sampling context. The methodology comprises a dynamic data structure that constructs a robust Bayesian model of the statistics generated by the heuristic process subject to an accuracy constraint, the posterior Kullback-Leibler divergence. Sampling from the dynamic structure will generally yield partial states, which are completed by recursively calling the heuristic to refine the structure and resuming the sampling. Our experiments on various Ising models suggest that state space sampling enables heuristic state generation with accurate probability estimates, demonstrated by illustrating the convergence of a simulated annealing process to the Boltzmann distribution with increasing run length. Consequently, heretofore unprecedented direct importance sampling using the emph{final} (marginal) distribution of a generic stochastic process is allowed, potentially augmenting the range of algorithms at the Monte Carlo practitioners disposal.
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