A note on semi-discrete conservation laws and conservation of wave action by multisymplectic Runge-Kutta box schemes

In this note we show that multisymplectic Runge-Kutta box schemes, of which the Gauss-Legendre methods are the most important, preserve a discrete conservation law of wave action. The result follows by loop integration over an ensemble of flow realizations, and the local energy-momentum conservation law for continuous variables in semi-discretization

Conservation of wave action under multisymplectic discretizations

In this paper we discuss the conservation of wave action under numerical discretization by variational and multisymplectic methods. Both the general wave action conservation defined with respect to a smooth, periodic, one-parameter ensemble of flow realizations and the specific wave action based on an approximated and averaged Lagrangian are addressed in the numerical context. It is found that the discrete variational formulation gives rise in a natural way not only to the discrete wave action conservation law but to a generalization of the numerical dispersion relation to the case of variable coefficients. Indeed a fully discrete analog of the modulation equations arises. On the other hand the multisymplectic framework gives easy access to the conservation law for the general class of multisymplectic Runge-Kutta methods. A numerical experiment confirms conservation of wave action to machine precision and suggests that the solution of the discrete modulation equations approximates the numerical solution to order O(e) on intervals of O(

On the construction of deflation-based preconditioners

In this article we introduce new bounds on the effective condition number of deflated and preconditioned-deflated symmetric positive definite linear systems. For the case of a subdomain deflation such as that of Nicolaides [SIAM J. Numer. Anal., 24 (1987), pp. 355--365], these theorems can provide direction in choosing a proper decomposition into subdomains. If grid refinement is performed, keeping the subdomain grid resolution fixed, the condition number is insensitive to the grid size. Subdomain deflation is very easy to implement and has been parallelized on a distributed memory system with only a small amount of additional communication. Numerical experiments for a steady-state convection-diffusion problem are included

The hamiltonian particle-mesh method for the spherical shallow water equations

The Hamiltonian particle-mesh (HPM) method is generalized to the spherical shallow water equations, utilizing constrained particle dynamics on the sphere and smoothing with Merilees' double-periodic FFT formulation of O(J2 log J) in the latitudinal gridsize. The time step for the explicit, symplectic integrator depends only on the uniform smoothing length

Statistical mechanics of Arakawa`s discretizations

The results of statistical analysis of simulation data obtained from long-time integrations of geophysical fluid models greatly depend on the conservation properties of the numerical discretization chosen. Statistical mechanical theories are constructed for three discretizations of the quasi-geostrophic model due to Arakawa (1966), each having different conservation properties. It is shown that the three statistical theories accurately explain the differences observed in statistics derived from the discretizations. The effect of the time discretization is also considered, and it is shown that projection is insufficient if the underlying spatial discretization is not conservative

Statistical relevance of vorticity conservation with the Hamiltonian particle-mesh method

We conduct long-time simulations with a Hamiltonian particle-mesh method for ideal fluid flow, to determine the statistical mean vorticity field of the discretization. Lagrangian and Eulerian statistical models are proposed for the discrete dynamics, and these are compared against numerical experiments. The observed results are in excellent agreement with the theoretical models, as well as with the continuum statistical mechanical theory for ideal fluid flow developed by Ellis et al. (2002). In particular the results verify that the apparently trivial conservation of potential vorticity along particle paths within the HPM method significantly influences the mean state. As a side note, the numerical experiments show that a nonzero fourth moment of potential vorticity can influence the statistical mean

10Gbit/s modulation of a fast switching slotted Fabry-Pérot tunable laser

The device used is a three-section, 3mum wide ridge waveguide laser based on commercially available material. During the fabrication a series of slots are introduced into the front and back sections, which act as sites of internal reflections. The slots are etched to a depth that just penetrates the top of the upper waveguide resulting in an internal reflectance of-1% at each slot. The front, middle, and back sections are 180, 690 and 170 microns long respectively. In this work the back and middle sections are tied together electrically allowing simpler control of the device. By varying the applied DC currents, eight discrete channels are observed over a range of approximately 19nm

Hydrostatic Hamiltonian particle-mesh (HPM) methods for atmospheric modelling

We develop a hydrostatic Hamiltonian particle-mesh (HPM) method for efficient long-term numerical integration of the atmosphere. In the HPM method, the hydrostatic approximation is interpreted as a holonomic constraint for the vertical position of particles. This can be viewed as defining a set of vertically buoyant horizontal meshes, with the altitude of each mesh point determined so as to satisfy the hydrostatic balance condition and with particles modelling horizontal advection between the moving meshes. We implement the method in a vertical-slice model and evaluate its performance for the simulation of idealized linear and nonlinear orographic flow in both dry and moist environments. The HPM method is able to capture the basic features of the gravity wave to a degree of accuracy comparable with that reported in the literature. The numerical solution in the moist experiment indicates that the influence of moisture on wave characteristics is represented reasonably well and the reduction of momentum flux is in good agreement with theoretical analysis. Copyright © 2011 Royal Meteorological Societ

Linear PDEs and numerical methods that preserve a multi-symplectic conservation law

Multisymplectic methods have recently been proposed as a generalization of symplectic ODE methods to the case of Hamiltonian PDEs. Their excellent long time behavior for a variety of Hamiltonian wave equations has been demonstrated in a number of numerical studies. A theoretical investigation and justification of multisymplectic methods is still largely missing. In this paper, we study linear multisymplectic PDEs and their discretization by means of numerical dispersion relations. It is found that multisymplectic methods in the sense of Bridges and Reich Phys. Lett. A, 284 (2001), pp. 184-193] and Reich J. Comput. Phys., 157 (2000), pp. 473-499], such as Gauss-Legendre Runge-Kutta methods, possess a number of desirable properties such as nonexistence of spurious roots and conservation of the sign of the group velocity. A certain CFL-type restriction on $\Delta t/\Delta x$ might be required for methods higher than second order in time. It is also demonstrated by means of the explicit midpoint method that multistep methods may exhibit spurious roots in the numerical dispersion relation for any value of $\Delta t/\Delta x$ despite being multisymplectic in the sense of discrete variational mechanics [J. E. Marsden, G. P. Patrick, and S. Shkoller, Commun. Math. Phys., 199 (1999), pp. 351-395]