Numerical solution of the one-dimensional Burgers’ equation: Implicit and fully implicit exponential finite difference methods

This paper describes two new techniques which give improved exponential finite difference solutions of Burgers’ equation. These techniques are called implicit exponential finite difference method and fully implicit exponential finite difference method for solving Burgers’ equation. As the Burgers’ equation is nonlinear, the scheme leads to a system of nonlinear equations. At each time-step, Newton’s method is used to solve this nonlinear system. The results are compared with exact values and it is clearly shown that results obtained using both the methods are precise and reliable.     

Analysis of numerical dispersion properties of SFDTD and MRTD schemes

In this paper, the Maxwell's equations are written as Hamilton canonical equations by using Hamilton functional variation method. Maxwell's equations can be discretized with symplectic propagation technique combined with high-order difference schemes approximations to construct symplectic finite difference time domain (SFDTD) method. The high-order dispersion equations of the scheme for space is deduced. The numerical dispersion analysis is included, and it is compared with the multiresolution time-domain (MRTD) method based on the Daubechies scaling functions. Numerical results show high efficiency and accuracy of the SFDTD method.     

气相爆轰高阶中心差分-WENO组合格式自适应网格方法

研究一种高阶中心差分-WENO组合格式,并采用自适应网格方法进行二维和三维气相爆轰波的数值模拟.采用ZND爆轰模型的控制方程为包含化学反应源项的Euler方程组.组合格式在大梯度区采用WENO格式捕捉间断,在光滑区采用高阶中心差分格式提高计算效率.采用一种基于流场结构特征的自适应网格.计算结果,表明这种方法同时具有高精度、高分辨率和高效率的特点.     

An element-free Galerkin(EFG) method for numerical solution of the coupled Schrdinger-KdV equations

The present paper deals with the numerical solution of the coupled Schrdinger-KdV equations using the elementfree Galerkin(EFG) method which is based on the moving least-square approximation.Instead of traditional mesh oriented methods such as the finite difference method(FDM) and the finite element method(FEM),this method needs only scattered nodes in the domain.For this scheme,a variational method is used to obtain discrete equations and the essential boundary conditions are enforced by the penalty method.In numerical experiments,the results are presented and compared with the findings of the finite element method,the radial basis functions method,and an analytical solution to confirm the good accuracy of the presented scheme.     

A diffuse-interface method for two-phase flows with soluble surfactants

A method is presented to solve two-phase problems involving soluble surfactants. The incompressible Navier-Stokes equations are solved along with equations for the bulk and interfacial surfactant concentrations. A non-linear equation of state is used to relate the surface tension to the interfacial surfactant concentration. The method is based on the use of a diffuse interface, which allows a simple implementation using standard finite difference or finite element techniques. Here, finite difference methods on a block-structured adaptive grid are used, and the resulting equations are solved using a non-linear multigrid method. Results are presented for a drop in shear flow in both 2D and 3D, and the effect of solubility is discussed.     

Pseudo-spectral methods and linear instabilities in reaction-diffusion fronts

We explore the application of a pseudo-spectral Fourier method to a set of reaction-diffusion equations and compare it with a second-order finite difference method. The prototype cubic autocatalytic reaction-diffusion model as discussed by Gray and Scott [Chem. Eng. Sci. 42, 307 (1987)] with a nonequilibrium constraint is adopted. In a spatial resolution study we find that the phase speeds of one-dimensional finite amplitude waves converge more rapidly for the spectral method than for the finite difference method. Furthermore, in two dimensions the symmetry preserving properties of the spectral method are shown to be superior to those of the finite difference method. In studies of plane/axisymmetric nonlinear waves a symmetry breaking linear instability is shown to occur and is one possible route for the formation of patterns from infinitesimal perturbations to finite amplitude waves in this set of reaction-diffusion equations. (c) 1996 American Institute of Physics.     

Two-Level Hierarchical FEM Method for Modeling Passive Microwave Devices

In recent years multigrid methods have been proven to be very efficient for solving large systems of linear equations resulting from the discretization of positive definite differential equations by either the finite difference method or theh-version of the finite element method. In this paper an iterative method of the multiple level type is proposed for solving systems of algebraic equations which arise from thep-version of the finite element analysis applied to indefinite problems. A two-levelV-cycle algorithm has been implemented and studied with a Gauss–Seidel iterative scheme used as a smoother. The convergence of the method has been investigated, and numerical results for a number of numerical examples are presented.     

流体力学方程的间断有限元方法

在二维区域三角形网格上应用一阶、二阶和三阶精度间断有限元方法,对流体力学方程和方程组进行了数值模拟.计算结果与差分方法计算结果比较,认为间断有限元方法在求解复杂边界条件和区域问题上有一定的优势.     

Solution of Hartree-Fock equations in coordinate space for axially symmetric nuclei

As an alternative to the conventional deformed harmonic oscillator basis expansion, a method is developed in which the coupled integro-differential Hartree-Fock equations are solved directly in coordinate space for a simple effective interaction. The single particle wave functions are obtained on a finite mesh by minimizing a discretized energy functional and solving the resulting finite difference equations using the Lanczos algorithm. Expressions to correct the total energy to second order in the mesh spacing are derived, and the accuracy of the method is demonstrated by numerical comparison with spherical results. Applications and advantages of this new technique are briefly discussed.     

A Finite Volume Method for the Approximation of Diffusion Operators on Distorted Meshes

A new finite volume method is presented for discretizing general linear or nonlinear elliptic second-order partial-differential equations with mixed boundary conditions. The advantage of this method is that arbitrary distorted meshes can be used without the numerical results being altered. The resulting algorithm has more unknowns than standard methods like finite difference or finite element methods. However, the matrices that need to be inverted are positive definite, so the most powerful linear solvers can be applied. The method has been tested on a few elliptic and parabolic equations, either linear, as in the case of the standard heat diffusion equation, or nonlinear, as in the case of the radiation diffusion equation and the resistive diffusion equation with Hall term.     

Numerical analysis of uncertain temperature field by stochastic finite difference method

Based on the combination of stochastic mathematics and conventional finite difference method,a new numerical computing technique named stochastic finite difference for solving heat conduction problems with random physical parameters,initial and boundary conditions is discussed.Begin with the analysis of steady-state heat conduction problems,difference discrete equations with random parameters are established,and then the computing formulas for the mean value and variance of temperature field are derived by the second-order stochastic parameter perturbation method.Subsequently,the proposed random model and method are extended to the field of transient heat conduction and the new analysis theory of stability applicable to stochastic difference schemes is developed.The layer-by-layer recursive equations for the first two probabilistic moments of the transient temperature field at different time points are quickly obtained and easily solved by programming.Finally,by comparing the results with traditional Monte Carlo simulation,two numerical examples are given to demonstrate the feasibility and effectiveness of the presented method for solving both steady-state and transient heat conduction problems.     

用基于磁场的改进型有限差分法分析光波导的全矢量本征模

提出一种改进的有限差分法,用以求解全矢量磁场波方程,分析光波导承载的全矢量本征模.离散交叉项时,采用与波导结构无关的六点差分格式,考虑磁场分量的导数在芯包分界处的不连续性,比传统四点差分格式有更高的计算精度.分析阶跃型光纤、矩形和脊形光波导的全矢量本征模问题,给出全矢量基模的磁场分布及其归一化传播常数,揭示全矢量本征模的混合特性,所得结果与解析法、高精度模横向谐振法的结果吻合,验证了方法的收敛性与计算精度.     

A Finite Variable Difference Relaxation Scheme for hyperbolic–parabolic equations

Using the framework of a new relaxation system, which converts a nonlinear viscous conservation law into a system of linear convection–diffusion equations with nonlinear source terms, a finite variable difference method is developed for nonlinear hyperbolic–parabolic equations. The basic idea is to formulate a finite volume method with an optimum spatial difference, using the Locally Exact Numerical Scheme (LENS), leading to a Finite Variable Difference Method as introduced by Sakai [Katsuhiro Sakai, A new finite variable difference method with application to locally exact numerical scheme, Journal of Computational Physics, 124 (1996) pp. 301–308.], for the linear convection–diffusion equations obtained by using a relaxation system. Source terms are treated with the well-balanced scheme of Jin [Shi Jin, A steady-state capturing method for hyperbolic systems with geometrical source terms, Mathematical Modeling Numerical Analysis, 35 (4) (2001) pp. 631–645]. Bench-mark test problems for scalar and vector conservation laws in one and two dimensions are solved using this new algorithm and the results demonstrate the efficiency of the scheme in capturing the flow features accurately.     

Numerical solutions to 2D Maxwell–Bloch equations

Rare-earth-doped crystals contain inhomogeneously broadened two-level atoms. Optical propagation and nonlinear interaction in the crystals can be described by the Maxwell–Bloch equations. We show a consistent numerical approach that solves Maxwell’s equations by using the FFT-finite difference beam propagation method and the Bloch equations by using the finite difference method. Numerical simulation results are given for an off-axis 3-pulse photon echo.     

基于SNGR方法的后缘噪声数值模拟

采用随机噪声产生和传播(SNGR)方法对后缘噪声进行数值模拟.SNGR方法结合随机方法和计算流体力学,耗费较少的计算资源就可以预测噪声水平.数值模拟时采用有限体积法求解雷诺平均Navier-Stokes(RANS)方程;采用有限差分法求解声学扰动方程,数值格式采用色散关系保持(DRP)格式,远场边界条件采用无反射边界条件.以二维平板和NACA0012翼型为例,编制程序,与参考结果对比表明,程序可以预测后缘噪声.     

高速碰撞数值计算中的光滑粒子法

扼要讨论了改进的光滑粒子法的离散思想,给出了二维轴对称问题中连续介质力学守恒方程的离散过程及离散格式,提出了轴对称坐标下确定影响域内粒子数的方法.最后通过高速碰撞的系列算例说明,光滑粒子法不但适宜于计算大变形冲击力学问题,而且有着其它网格法所无法替代的优势.     

Numerical solution of the conformable space-time fractional wave equation

In this paper, an efficient numerical method is considered for solving space-time fractional wave equation. The fractional derivatives are described in the conformable sense. The method is based on shifted Chebyshev polynomials of the second kind. Unknown function is written as Chebyshev series with the N term. The space-time fractional wave equation is reduced to a system of ordinary differential equations by using the properties of Chebyshev polynomials. The finite difference method is applied to solve this system of equations. Numerical results are provided to verify the accuracy and efficiency of the proposed approach.     

A New Finite Volume Element Formulation for the Non-Stationary Navier-Stokes Equations

A semi-discrete scheme about time for the non-stationary Navier-Stokes equations is presented firstly, then a new fully discrete finite volume element (FVE) formulation based on macroelement is directly established from the semi-discrete scheme about time. And the error estimates for the fully discrete FVE solutions are derived by means of the technique of the standard finite element method. It is shown by numerical experiments that the numerical results are consistent with theoretical conclusions. Moreover, it is shown that the FVE method is feasible and efficient for finding the numerical solutions of the non-stationary Navier-Stokes equations and it is one of the most effective numerical methods among the FVE formulation, the finite element formulation, and the finite difference scheme.     

Collocation Methods for Hyperbolic Partial Differential Equations with Singular Sources

A numerical study is given on the spectral methods and the high order WENO finite difference scheme for the solution of linear and nonlinear hyperbolic partial differential equations with stationary and non-stationary singular sources. The singular source term is represented by the $δ$-function. For the approximation of the $δ$-function, the direct projection method is used that was proposed in [6]. The $δ$-function is constructed in a consistent way to the derivative operator. Nonlinear sine-Gordon equation with a stationary singular source was solved with the Chebyshev collocation method. The $δ$-function with the spectral method is highly oscillatory but yields good results with small number of collocation points. The results are compared with those computed by the second order finite difference method. In modeling general hyperbolic equations with a non-stationary singular source, however, the solution of the linear scalar wave equation with the non-stationary singular source using the direct projection method yields non-physical oscillations for both the spectral method and the WENO scheme. The numerical artifacts arising when the non-stationary singular source term is considered on the discrete grids are explained.     

高功率微波与等离子体相互作用理论和数值研究

研究高功率微波与等离子体的相互作用,对于微波放电和电磁兼容研究均具有重要意义.基于波动方程、等离子体的流体力学方程以及波尔兹曼方程,建立高功率微波脉冲与等离子体相互作用的理论模型,并结合等离子体的特征参数,采用时域有限差分方法分析了等离子体电子密度和高功率微波传输特性的变化.结果表明,由于高功率微波的电子加热作用,等离子体中的非线性效应明显,发生击穿使得等离子体电子密度增大,从而导致微波的反射增强,透过率降低.所提出的模型和相关结果对于高功率微波和电磁脉冲防护具有指导意义.