The Best Finite-Difference Scheme for the Helmholtz Equation

The best finite-difference scheme for the Helmholtz equation is suggested. A method of solving obtained finite-difference scheme is developed. The efficiency and accuracy of method were tested on several examples.

AN EXPONENTIALLY FITTED DIFFERENCE SCHEME FOR THE HYPERBOLIC-HYPERBOLIC SINGULARLY PERTURBED INITIAL-BOUNDARY VALUE PROBLEM

In this paper we discuss, an initial-boundary value problem of hyperbolic type with first derivative with respect to x. The asymptotic solution is constructed and its uniform validity is proved under weader compatibility conditions. Then we develop an exponentially fitted difference scheme and establish discrete energy inequality. Finally, we prove that the solution of difference problem uniformly converges to the solution of the original problem.

Optimization of a global seventh-order dissipative compact finite-difference scheme by a genetic algorithm

A global seventh-order dissipative compact finite-difference scheme is optimized in terms of time stability. The dissipative parameters appearing in the boundary closures are assumed to be different, resulting in an optimization problem with several parameters determined by applying a generic algorithm. The optimized schemes are analyzed carefully from the aspects of the eigenvalue distribution, the ε-pseudospectra, the short time behavior, and the Fourier analysis. Numerical experiments for the Euler equations are used to show the effectiveness of the final recommended scheme.

Perfect plane-wave source for a high-order symplectic finite-difference time-domain scheme

The method of splitting a plane-wave finite-difference time-domain （SP-FDTD） algorithm is presented for the initiation of plane-wave source in the total-field / scattered-field （TF/SF） formulation of high-order symplectic finite- difference time-domain （SFDTD） scheme for the first time. By splitting the fields on one-dimensional grid and using the nature of numerical plane-wave in finite-difference time-domain （FDTD）, the identical dispersion relation can be obtained and proved between the one-dimensional and three-dimensional grids. An efficient plane-wave source is simulated on one-dimensional grid and a perfect match can be achieved for a plane-wave propagating at any angle forming an integer grid cell ratio. Numerical simulations show that the method is valid for SFDTD and the residual field in SF region is shrinked down to -300 dB.

An efficient locally one-dimensional finite-difference time-domain method based on the conformal scheme

An efficient conformal locally one-dimensional finite-difference time-domain（LOD-CFDTD） method is presented for solving two-dimensional（2D） electromagnetic（EM） scattering problems. The formulation for the 2D transverse-electric（TE） case is presented and its stability property and numerical dispersion relationship are theoretically investigated. It is shown that the introduction of irregular grids will not damage the numerical stability. Instead of the staircasing approximation, the conformal scheme is only employed to model the curve boundaries, whereas the standard Yee grids are used for the remaining regions. As the irregular grids account for a very small percentage of the total space grids, the conformal scheme has little effect on the numerical dispersion. Moreover, the proposed method, which requires fewer arithmetic operations than the alternating-direction-implicit（ADI） CFDTD method, leads to a further reduction of the CPU time. With the total-field/scattered-field（TF/SF） boundary and the perfectly matched layer（PML）, the radar cross section（RCS） of two2 D structures is calculated. The numerical examples verify the accuracy and efficiency of the proposed method.

Characteristic Analysis of Exponential Compact Higher Order Schemes for Convection-Diffusion Equations

This paper looks at the development of a class of Exponential Compact Higher Order (ECHO) schemes and attempts to comprehend their behaviour by introducing different combinations of discrete source function and its derivatives. The characteristic analysis is performed for one-dimensional schemes to understand the efficiency of the scheme and a similar analysis has been introduced for higher dimensional schemes. Finally, the developed schemes are used to solve several example problems and compared the error norms and rates of convergence.

SEVERAL NEW TYPES OF FINITE-DIFFERENCE SCHEMES FOR SHALLOW-WATER EQUATION WITH INITIAL-BOUNDARY VALUE AND THEIR NUMERICAL EXPERIMENT

This paper proposed several new types of finite-difference methods for the shallow water equation in absolute coordinate system and put forward an effective two-step predictor-corrector method, a compact and iterative algorithm for five diagonal matrix. Then the iterative method was used for a multi-grid procedure for shallow water equation. A t last, an initial-boundary value problem was considered, and the numerical results show that the linear sinusoidal wave would successively evolve into conoidal wave.

Exponential time differencing based efficient SC-PML for RCS simulation

To efficiently simulate and calculate the radar cross section(RCS) related electromagnetic problems by employing the finite-difference time-domain(FDTD) algorithm, an efficient stretched coordinate perfectly matched layer(ESC-PML) based upon the exponential time differencing(ETD) method is proposed.The proposed implementation can not only reduce the number of auxiliary variables in the SC-PML regions but also maintain the ability of the original SC-PML in terms of the absorbing performance. Compared with the other existed algorithms, the ETDFDTD method shows the least memory consumption resulting in the computational efficiency. The effectiveness and efficiency of the proposed ESC-PML scheme is verified through the RCS relevant problems including the perfect E conductor(PEC) sphere model and the patch antenna model. The results indicate that the proposed scheme has the advantages of the ETD-FDTD method and ESC-PML scheme in terms of high computational efficiency and considerable computational accuracy.

Finite-difference modeling with variable grid-size and adaptive time-step in porous media

Forward modeling of elastic wave propagation in porous media has great importance for understanding and interpreting the influences of rock properties on characteristics of seismic wavefield. However,the finite-difference forward-modeling method is usually implemented with global spatial grid-size and time-step; it consumes large amounts of computational cost when small-scaled oil/gas-bearing structures or large velocity-contrast exist underground. To overcome this handicap,combined with variable grid-size and time-step,this paper developed a staggered-grid finite-difference scheme for elastic wave modeling in porous media. Variable finite-difference coefficients and wavefield interpolation were used to realize the transition of wave propagation between regions of different grid-size. The accuracy and efficiency of the algorithm were shown by numerical examples. The proposed method is advanced with low computational cost in elastic wave simulation for heterogeneous oil/gas reservoirs.

DIFFERENCE SCHEME FOR AN INITIAL-BOUNDARY VALUE PROBLEM FOR LINEAR COEFFICIENT-VARIED PARABOLIC DIFFERENTIAL EQUATION WITH A NONSMOOTH BOUNDARY LAYER FUNCTION

In this paper, using nonuniform mesh and exponentially fitted difference method, a uniformly convergent difference scheme for an initial-boundary value problem of linear parabolic differential equation with the nonsmooth boundary layer function with respect to small parameter e is given, and error estimate and numerical result are also given.

High Order of Accuracy for Poisson Equation Obtained by Grouping of Repeated Richardson Extrapolation with Fourth Order Schemes

In this article,we improve the order of precision of the two-dimensional Poisson equation by combining extrapolation techniques with high order schemes.The high order solutions obtained traditionally generate non-sparse matrices and the calculation time is very high.We can obtain sparse matrices by applying compact schemes.In this article,we compare compact and exponential finite difference schemes of fourth order.The numerical solutions are calculated in quadruple precision(Real*16 or extended precision)in FORTRAN language,and iteratively obtained until reaching the round-off error magnitude around 1.0E−32.This procedure is performed to ensure that there is no iteration error.The Repeated Richardson Extrapolation(RRE)method combines numerical solutions in different grids,determining higher orders of accuracy.The main contribution of this work is based on a process that initializes with fourth order solutions combining with RRE in order to find solutions of sixth,eighth,and tenth order of precision.The multigrid Full Approximation Scheme(FAS)is also applied to accelerate the convergence and obtain the numerical solutions on the fine grids.

Bayes Prediction of Future Observables from Exponentiated Populations with Fixed and Random Sample Size

Bayesian predictive probability density function is obtained when the underlying pop-ulation distribution is exponentiated and subjective prior is used. The corresponding predictive survival function is then obtained and used in constructing 100(1 – ?)% predictive interval, using one- and two- sample schemes when the size of the future sample is fixed and random. In the random case, the size of the future sample is assumed to follow the truncated Poisson distribution with parameter λ. Special attention is paid to the exponentiated Burr type XII population, from which the data are drawn. Two illustrative examples are given, one of which uses simulated data and the other uses data that represent the breaking strength of 64 single carbon fibers of length 10, found in Lawless [40].

Statistical Inference of Stress-Strength Model for Exponentiated Pareto Model with Censored Data

The reliability of a system is discussed when the strength of the system and the stress imposed on it are independent and non-identical exponentiated Pareto distributed random variables with progressively censored scheme.Different interval estimations are proposed.The interval estimations obtained are exact,approximate and bootstrap confidence intervals.Different methods and the corresponding confidence intervals are compared using Monte-Carlo simulations.Simulation results show that the confidence intervals(CIs)of exact and approximate methods are really better than those of the bootstrap method.

A Numerical Solution of Heat Equation for Several Thermal Diffusivity Using Finite Difference Scheme with Stability Conditions

The heat equation is a second-order parabolic partial differential equation, which can be solved in many ways using numerical methods. This paper provides a numerical solution that uses the finite difference method like the explicit center difference method. The forward time and centered space (FTCS) is used to a problem containing the one-dimensional heat equation and the stability condition of the scheme is reported with different thermal conductivity of different materials. In this study, results obtained for different thermal conductivity of distinct materials are compared. Also, the results reveal the well-behavior properties of the materials in good agreement.

Overview of computation strategies on the dispersion analysis for explicit finite difference solution of acoustic wave equation

Finite-difference(FD)method is the most extensively employed numerical modeling technique.Nevertheless,when using the FD method to simulate the seismic wave propagation,the large spatial or temporal sampling interval can lead to dispersion errors and numerical instability.In the FD scheme,the key factor in determining both dispersion errors and stability is the selection of the FD weights.Thus,How to obtain appropriate FD weights to guarantee a stable numerical modeling process with minimum dispersion error is critical.The FD weights computation strategies can be classified into three types based on different computational ideologies,window function strategy,optimization strategy,and Taylor expansion strategy.In this paper,we provide a comprehensive overview of these three strategies by presenting their fundamental theories.We conduct a set of comparative analyses of their strengths and weaknesses through various analysis tests and numerical modelings.According to these comparisons,we provide two potential research directions of this field:Firstly,the development of a computational strategy for FD weights that enhances stability;Secondly,obtaining FD weights that exhibit a wide bandwidth while minimizing dispersion errors.

Nonstandard finite-difference schemes for the two-level Bloch model

In this paper, we present splitting schemes for the two-level Bloch model. After proposing two ways to split the Bloch equation, we show that it is possible in each case togenerate exact numerical solutions of the obtained sub-equations. These exact solutionsinvolve matrix exponentials which can be expensive to compute. Here, for 2×2 matriceswe develop equivalent formulations which reduce the computational cost. These splittingschemes are nonstandard ones and conserve all the physical properties (Hermicity, positiveness and trace) of Bloch equations. In addition, they are explicit, making effectivetheir implementation when coupled with the Maxwell’s equations.

EXPONENTIALLY FITTED TRAPEZOIDAL SCHEME FOR A STOCHASTIC OSCILLATOR

This paper applies exponentially fitted trapezoidal scheme to a stochastic oscillator. The scheme is convergent with mean-square order 1 and symplectic. Its numerical solution oscillates and the second moment increases linearly with time. The numerical example verifies the analysis of the scheme.

The approach to optimization of finite-difference schemes for the advective stage of finite-difference-based lattice Boltzmann method

The approach to optimization of finite-difference(FD)schemes for the linear advection equation(LAE)is proposed.The FD schemes dependent on the scalar dimensionless parameter are considered.The parameter is included in the expression,which approximates the term with spatial derivatives.The approach is based on the considering of the dispersive and dissipative characteristics of the schemes as the functions of the parameter.For the proper choice of the parameter,these functions are minimized.The approach is applied to the optimization of two-step schemes with an asymmetric approximation of time derivative and with various approximations of the spatial term.The cases of schemes from first to fourth approximation orders are considered.The optimal values of the parameter are obtained.Schemes with the optimal values are applied to the solution of test problems with smooth and discontinuous initial conditions.Also,schemes are used in the FD-based lattice Boltzmann method(LBM)for modeling of the compressible gas flow.The obtained numerical results demonstrate the convergence of the schemes and decaying of the numerical dispersion.

四阶指数差分及其在FDTD中的应用

提出一种新的指数差分格式。与普通的二阶中心差分格式相比,该格式具有在不增加存储量的前提下提高计算精度的优点。文中用实例验证了该差分格式的高精度性。最后,应用该方法计算了圆柱凹面反射的问题,得出凹面内场的分布图。

对流扩散方程的四阶指数型差分格式

本文提出差分格式的摄动方法,对二阶指数型格式中对流系数和源项作二阶修正,推演出对流扩散方程的四阶指数型格式。该四阶格式的基本结构与二阶指数型格式完全相同,且其系数或源项中所含二阶修正可根据二阶格式计算结果一次性确定,使得计算十分简便。一至三维的四阶指数型格式均具有无条件稳定性,用于Burgers方程等流体力学模型问题,且与常用格式进行了比较,显示出良好的精度,并能较好地适应大梯度区域。