Multi-moment advection scheme for Vlasov simulations

We present a new numerical scheme for solving the advection equation and its application to Vlasov simulations. The scheme treats not only point values of a profile but also its zeroth to second order piecewise moments as dependent variables, for better conservation of the information entropy. We have developed one-and two-dimensional schemes and show that they provide quite accurate solutions within reasonable usage of computational resources compared to other existing schemes. The two-dimensional scheme can accurately solve the solid body rotation problem of a gaussian profile for more than hundred rotation periods with little numerical diffusion. This is crucially important for Vlasov simulations of magnetized plasmas. Applications of the one- and two-dimensional schemes to electrostatic and electromagnetic Vlasov simulations are presented with some benchmark tests.     

On simplifying ‘incremental remap’–based transport schemes

The flux-form incremental remapping transport scheme introduced by Dukowicz and Baumgardner [1] converts the transport problem into a remapping problem. This involves identifying overlap areas between quadrilateral flux-areas and regular square grid cells which is non-trivial and leads to some algorithm complexity. In the simpler swept area approach (originally introduced by Hirt et al. [2]) the search for overlap areas is eliminated even if the flux-areas overlap several regular grid cells. The resulting simplified scheme leads to a much simpler and robust algorithm.     

A semilocal convergence result for Newton’s method under generalized conditions of Kantorovich

From Kantorovich’s theory we establish a general semilocal convergence result for Newton’s method based fundamentally on a generalization required to the second derivative of the operator involved. As a consequence, we obtain a modification of the domain of starting points for Newton’s method and improve the a priori error estimates. Finally, we illustrate our study with an application to a special case of conservative problems.     

Flutter instability and other singularity phenomena in symmetric systems via combination of mass distribution and weak damping

The local dynamic instability of autonomous conservative, lumped-mass (discrete) systems, is thoroughly discussed when negligibly small dissipative forces are included. It is shown that such small forces may change drastically the response of these systems. Hence, existing, widely accepted, findings based on the omission of damping could not be valid if damping, being always present in actual systems, is included. More specifically the conditions under which the above systems may experience dynamic bifurcations associated either with a degenerate or a generic Hopf bifurcation are examined in detail by studying the effect of the structure of the damping matrix on the Jacobian eigenvalues. The case whereby this phenomenon may occur before divergence is discussed in connection with the individual or coupling effect of non-uniform mass and stiffness distribution. Jump phenomena in the critical dynamic loading at a certain mass distribution are also assessed. Numerical results verified by a non-linear dynamic analysis using 2-DOF and 3-DOF models confirm the validity of the theoretical findings as well as the efficiency of the technique proposed herein.     

A Trigger of Coupled Singularities

By using example of nonlinear dynamics of a pair of coupled gears, the phenomenon of appearance and disappearance of a trigger of coupled singularities and homoclinic orbits in the form of number eight in the phase portrait in the phase plane is investigated. That phenomenon is an accompanying phenomenon of loss of stability of the local unique equilibrium position. For a generalized case under certain conditions, a theorem of the appearance of a trigger of coupled singularities in a nonlinear dynamical conservative system, the first derivative of the system potential energy which is a product of two periodic functions with different periods, and one bifurcation parameter, which is the cause for the appearance of new roots of these two functions, is defined.     

Some remarks about the resolution of high velocity flows near low densities

High velocity flows which are exposed to strong rarefaction waves and creating low densities regions in it present difficulties and inaccuracies for numerical resolution. In particular, variables related to the internal energy are wrongly evaluated. Use of classical schemes solving the Euler equations in conservative variables introduces significant errors in the determination of temperature. We recommend to employ a non-conservative formulation of the energy equation. Results found to be more accurate in using the present internal energy formulation. In order to have the formulation available for both shock and strong rarefaction waves, we propose a hybrid formulation of conservative and non-conservative ones, depending on a shock indicator. The results are compared with exact solutions and show a significant improvement of the accuracy. The method is then extended to two-dimensional cases. Received 28 March 1997 / Accepted 18 June 1997     

Optical logic redux

Twenty years ago IBM physicist Robert Keyes published a paper entitled “Optical Logic—in the light of computer technology.” It caused an instant furor in the fledgling optical logic community. Now, 20 years after that devastating critique, the field of optical logic has grown enormously. There are literally thousands of papers. Many of them are collected in a bibliography given here. Was Keyes’ critique wrong? Have opticists simply ignored what Keyes pointed out? Have new developments made some of his remarks not quite so relevant? We argue here that

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Keyes was and still is mostly correct, but that may change in a few years

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Many researchers have indeed simply ignored what he said

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New developments in both optical logic and its applications open niches for optical logic that Keyes did not (and probably could not) anticipate

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New and anticipated developments in electronics may increase the role for optics

     

Upwind-biased FORCE schemes with applications to free-surface shallow flows

In this paper we develop numerical fluxes of the centred type for one-step schemes in conservative form for solving general systems of conservation laws in multiple-space dimensions on structured meshes. The proposed method is an extension of the multidimensional FORCE flux developed by Toro et al. (2009) [14]. Here we introduce upwind bias by modifying the shape of the staggered mesh of the original FORCE method. The upwind bias is evaluated using an estimate of the largest eigenvalue, which in any case is needed for selecting a time step. The resulting basic flux is first-order accurate and monotone. For the linear advection equation, the proposed UFORCE method reproduces exactly the upwind Godunov method. Extension to non-linear systems has been done empirically via the two-dimensional inviscid shallow water equations. Second order of accuracy in space and time on structured meshes is obtained in the framework of finite volume methods. The proposed method improves the accuracy of the solution for small Courant numbers and intermediate waves associated with linearly degenerate fields (contact discontinuities, shear waves and material interfaces). It achieves comparable accuracy to that of upwind methods with approximate Riemann solvers, though retaining the simplicity and efficiency of centred methods. The performance of the schemes is assessed on a suite of test problems for the two-dimensional shallow water equations.     

A conservative parallel difference method for 2-dimension diffusion equation

In this paper, a conservative parallel difference scheme, which is based on domain decomposition method, for 2-dimension diffusion equation is proposed. In the construction of this scheme, we use the numerical solution on the previous time step to give a weighted approximation of the numerical flux. Then the sub-problems with Neumann boundary are computed by fully implicit scheme. What is more, only local message communication is needed in the program. We use the method of discrete functional analysis to give the proof of the unconditional stability and second-order convergence accuracy. Some numerical tests are given to verify the theory results.