A method to simulate linear stability of impulsively accelerated density interfaces in ideal-MHD and gas dynamics

We present a numerical method to solve the linear stability of impulsively accelerated density interfaces in two dimensions such as those arising in the Richtmyer–Meshkov instability. The method uses an Eulerian approach, and is based on an upwind method to compute the temporally evolving base state and a flux vector splitting method for the perturbations. The method is applicable to either gas dynamics or magnetohydrodynamics. Numerical examples are presented for cases in which a hydrodynamic shock interacts with a single or double density interface, and a doubly shocked single density interface. Convergence tests show that the method is spatially second-order accurate for smooth flows, and between first and second-order accurate for flows with shocks.     

Analytic continuation of harmonic sums

We present a method for calculating any (nested) harmonic sum to arbitrary accuracy for all complex values of the argument. The method utilizes the relation between harmonic sums and (derivatives of) Hurwitz zeta functions, which allows a harmonic sum to be calculated as an expansion valid for large values of its argument. A program for implementing this method is also provided.     

The two-level element free Galerkin method for MHD flow at high Hartmann numbers

In this Letter, a new element free Galerkin method, namely the two-level element free Galerkin method, is presented for solving the governing equations of steady magnetohydrodynamic duct flow. Because this element free Galerkin method makes use of the nodal point configurations which do not require a mesh, therefore it differs from FEM-like approaches by avoiding the need of meshing, a very demanding task for complicated geometry problems. Another distinguished feature of the proposed method is the resolving capability of high gradients near the layer regions without local or adaptive refinements. Numerical results indicate that no matter how large the Hartmann number is, this method has the ability to produce the satisfactory results for the velocity and the magnetic field simultaneously. That is to say, the presented method has some excellent properties, such as better stability and accuracy.     

Solution of singular two-point boundary value problems using differential transformation method

In this Letter, we implemented relatively new, exact series method of solution known as the differential transform method for solving singular two-point boundary value problems. Several illustrative examples are given to demonstrate the effectiveness of the present method.     

The transfer-matrix and max-plus algebra method for global combinatorial optimization: Application to cyclic and polyhedral water clusters

A new method for the combinatorial optimization of quasi-one-dimensional systems is presented. This method is in close analogy with the well-known transfer-matrix method. The method allows for the calculation of the lowest energy levels of the system. However, when finding the ground and some low-lying states of large complex systems, this method is more economical when compared to the standard transfer-matrix method. The method presented here is based on max-plus algebra, which has maximization and addition as its basic arithmetic operations. For the explanation of this method we use cyclic water clusters as simple examples. The efficiency of the max-plus-algebraic method is demonstrated in the course of global combinatorial optimization of hydrogen bond arrangements in large polyhedral water clusters with fixed positions of the oxygen atoms. The energy of the clusters is estimated using approximate discrete models for the intermolecular interactions.     

Phase transitions in Edwards-Anderson model by means of information theory

A numeric method to characterize phase transitions is presented, explained and applied to a two-dimensional disordered system that can be thought of as a diluted ferromagnet or an Edwards-Anderson model near the ferromagnetic limit. A computer simulation is implemented to define a time series for order parameters; a file stores the time evolution of each parameter for different dilution concentrations and for a series of temperatures. These files are compressed and they reach a maximum size for temperatures in agreement with critical temperatures for the ferromagnetic/paramagnetic transition obtained by other methods. Site order parameter gives optimum results for this method based on data compression. Data compression procedures are invoked to give a qualitative explanation of this phenomenon. The advantages of this method are discussed by comparing results and procedures with two established methods: the crossing of Binder cumulants and the crossing of time autocorrelation functions. Other possible applications and extensions of this method are also mentioned.     

A Novel Method for Enhancing Goos-Hänchen Shift in Total Internal Reflection

Due to the tiny shift in order of optical wavelength for Goos-Hǎnchen （GH） shift, it is very difficult to directly measure and apply the GH shift. We develop a new method for enhancing GH shift of both TE and TM polarized waves. The method is based on a total reflection prism made of BK9 glass combined with a precise measurement of the resulting spatial displacement with a one-dimensional charge coupled device （CCD）. Measurements are performed to examine the validity of the method. Experimental and theoretical results indicate the feasibility of the method with an enhancement in optical wavelenghth shift at millimetre scale. The method is advantageous to application the GH shift in the optical domain, and is also meaningful for measuring even smaller changes in the refractive index of a liquid.     

Approximated Vector Diffraction Theory for Readout Signal of Multi-Level Run-Length-Limited Discs

Multi-level run-length-limited read-only optical storage is a kind of high density storage method. The width and height of recording marks are varied with different levels, which is the key technology for the optical storage. Then the readout signal of these discs with complex recording marks is computed by vector analysis method, it is very hard and time-consuming. Approximated vector computation combines the convenience of scalar method and precision of vector method, which is effective for multi-level run-length-limited read-only optical storage.     

A novel approach for solving the Fisher equation using Exp-function method

In this Letter, Exp-function method is employed to obtain traveling wave solutions of the Fisher equation. It is shown that, on this example, the Exp-function method is easy to implement and concise method for nonlinear evolution equations in mathematical physics.     

An evolutional approach to the numerical construction of polycrystalline structures using the Voronoi tessellation

We developed a method to numerically construct a polycrystalline structure with a specified grain-size distribution by utilizing the genetic algorithm, which is suitable for global optimization of complex parametric space, and the Voronoi tessellation. The computational merits of our new method compared with the conventional optimization method are: (a) it is able to escape more easily from local minima of the penalty function; and (b) it is suitable for parallel processing. The method efficiently creates input data for numerical simulations of microstructural evolution such as recrystallization, grain growth, deformation, and fracture.     

Inverse Scattering Method and Soliton Solution Family for String Effective Action

A modified Hauser-Ernst-type linear system is established and used to develop an inverse scattering method for solving the motion equations of the string effective action describing the coupled gravity, dilaton and KalbRamond fields. The reduction procedures in this inverse scattering method are found to be fairly simple, which makes the proposed inverse scattering method applied fine and effective. As an application, a concrete family of soliton solutions for the considered theory is obtained.     

Numerical solution of 3D Green's function for the dynamic system of anisotropic elasticity

In this Letter, we used homotopy perturbation method to obtain numerical solution of the 3D Green's function for the dynamic system of anisotropic elasticity. Application of homotopy perturbation method to this problem shows the rapid convergence of the sequence constructed by this method to the exact solution. The numerical results obtained from convolution of Green's function and data of the Cauchy problem are compared with the exact solution for cubic media. The results reveal that the proposed method is very effective and simple.     

Depolarization of multiply scattered light in transmission through a turbid medium with large particles

An approximate analytical method for solving the vector radiative transfer equation is proposed. The method is based on the assumption that single scattering of light by large-scale inhomogeneities occurs predominantly through small angles. The method is applied to calculate the polarization state of multiply scattered light. The results obtained are discussed for various turbid media.     

Display method with compensation of the spatial frequency response of a liquid crystal spatial light modulator for holographic femtosecond laser processing

Liquid crystal spatial light modulators, which are widely used as display devices for computer-generated holograms, have modulation characteristics that depend on spatial frequency. We describe a method for displaying a computer-generated hologram on a liquid crystal spatial light modulator with compensation of its spatial frequency response. Using this method, we demonstrate a binary phase grating with smaller dependence on the spatial frequency. We also demonstrate application of the display method to holographic femtosecond laser processing.     

Computation of derivatives of the rotation number for parametric families of circle diffeomorphisms

In this paper we present a numerical method to compute derivatives of the rotation number for parametric families of circle diffeomorphisms with high accuracy. Our methodology is an extension of a recently developed approach to compute rotation numbers based on suitable averages of iterates of the map and Richardson extrapolation. We focus on analytic circle diffeomorphisms, but the method also works if the maps are differentiable enough. In order to justify the method, we also require the family of maps to be differentiable with respect to the parameters and the rotation number to be Diophantine. In particular, the method turns out to be very efficient for computing Taylor expansions of Arnold Tongues of families of circle maps. Finally, we adapt these ideas to study invariant curves for parametric families of planar twist maps.     

Self-similar factor approximants for evolution equations and boundary-value problems

The method of self-similar factor approximants is shown to be very convenient for solving different evolution equations and boundary-value problems typical of physical applications. The method is general and simple, being a straightforward two-step procedure. First, the solution to an equation is represented as an asymptotic series in powers of a variable. Second, the series are summed by means of the self-similar factor approximants. The obtained expressions provide highly accurate approximate solutions to the considered equations. In some cases, it is even possible to reconstruct exact solutions for the whole region of variables, starting from asymptotic series for small variables. This can become possible even when the solution is a transcendental function. The method is shown to be more simple and accurate than different variants of perturbation theory with respect to small parameters, being applicable even when these parameters are large. The generality and accuracy of the method are illustrated by a number of evolution equations as well as boundary value problems.     

Two new solutions to the third-order symplectic integration method

Two new solutions are obtained for the symplecticity conditions of explicit third-order partitioned Runge-Kutta time integration method. One of them has larger stability limit and better dispersion property than the Ruth's method.     

Compact-2D FDTD for waveguides including materials with negative dielectric permittivity, magnetic permeability and refractive index

An efficient compact-2D finite-difference time-domain method is presented for the numerical analysis of guided modes in waveguides that may include negative dielectric permittivity, negative magnetic permeability and negative refractive index materials. Both complex variable and real variable methods are given. The method is demonstrated for the analysis of channel-plasmon-polariton guided modes in triangular groves on a metal surface. The presented method can be used for a range of waveguide problems that were previously unsolvable analytically, due to complex geometries, or numerically, due to computational requirements of conventional three-dimensional finite-difference time-domain methods. A three-dimensional finite-difference time-domain algorithm that also allows analysis in the presence of bound or free electric and equivalent magnetic charges is presented and an example negative refraction demonstrates the method.     

Hybrid Method for Tokamak MHD Equilibrium Configuration Reconstruction

A hybrid method for tokamak MHD equilibrium configuration reconstruction is proposed and employed in the modified EFIT code. This method uses the free boundary tokamak equilibrium configuration reconstruction algorithm with one boundary point fixed. The results show that the position of the fixed point has explicit effects on the reconstructed divertor configurations. In particular, the separatrix of the reconstructed divertor configuration precisely passes the required position when the hybrid method is used in the reconstruction. The profiles of plasma parameters such as pressure and safety factor for reconstructed HL-2A tokamak configurations with the hybrid and the free boundary methods are compared. The possibility for applications of the method to swing the separatrix strike point on the divertor target plate is discussed.     

Generalized differential transform method to differential-difference equation

In this Letter, we generalize the differential transform method to solve differential-difference equation for the first time. Two simple but typical examples are applied to illustrate the validity and the great potential of the generalized differential transform method in solving differential-difference equation. A Padé technique is also introduced and combined with GDTM in aim of extending the convergence area of presented series solutions. Comparisons are made between the results of the proposed method and exact solutions. Then we apply the differential transform method to the discrete KdV equation and the discrete mKdV equation, and successfully obtain solitary wave solutions. The results reveal that the proposed method is very effective and simple. We should point out that generalized differential transform method is also easy to be applied to other nonlinear differential-difference equation.