Subdomain Chebyshev Spectral Method for 2D and 3D Numerical Differentiations in a Curved Coordinate System

A new numerical approach, called the “subdomain Chebyshev spectral method” is presented for calculation of the spatial derivatives in a curved coordinate system, which may be employed for numerical solutions of partial differential equations defined in a 2D or 3D geological model. The new approach refers to a “strong version” against the “weak version” of the subspace spectral method based on the variational principle or Galerkin’s weighting scheme. We incorporate local nonlinear transformations and global spline interpolations in a curved coordinate system and make the discrete grid exactly matches geometry of the model so that it is achieved to convert the global domain into subdomains and apply Chebyshev points to locally sampling physical quantities and globally computing the spatial derivatives. This new approach not only remains exponential convergence of the standard spectral method in subdomains, but also yields a sparse assembled matrix when applied for the global domain simulations. We conducted 2D and 3D synthetic experiments and compared accuracies of the numerical differentiations with traditional finite difference approaches. The results show that as the points of differentiation vector are larger than five, the subdomain Chebyshev spectral method significantly improve the accuracies of the finite difference approaches.

Efficient Chebyshev Spectral Method for Solving Linear Elliptic PDEs Using Quasi-Inverse Technique

We present a systematic and efficient Chebyshev spectral method using quasiinverse technique to directly solve the second order equation with the homogeneous Robin boundary conditions and the fourth order equation with the first and second boundary conditions.The key to the efficiency of the method is to multiply quasiinverse matrix on both sides of discrete systems,which leads to band structure systems.We can obtain high order accuracy with less computational cost.For multi-dimensional and more complicated linear elliptic PDEs,the advantage of this methodology is obvious.Numerical results indicate that the spectral accuracy is achieved and the proposed method is very efficient for 2-D high order problems.

Dynamic analysis of beam-cable coupled systems using Chebyshev spectral element method

The dynamic characteristics of a beam–cable coupled system are investigated using an improved Chebyshev spectral element method in order to observe the effects of adding cables on the beam. The system is modeled as a double Timoshenko beam system interconnected by discrete springs. Utilizing Chebyshev series expansion and meshing the system according to the locations of its connections,numerical results of the natural frequencies and mode shapes are obtained using only a few elements, and the results are validated by comparing them with the results of a finiteelement method. Then the effects of the cable parameters and layout of connections on the natural frequencies and mode shapes of a fixed-pinned beam are studied. The results show that the modes of a beam–cable coupled system can be classified into two types, beam mode and cable mode, according to the dominant deformation. To avoid undesirable vibrations of the cable, its parameters should be controlled in a reasonable range, or the layout of the connections should be optimized.

A lumped mass Chebyshev spectral element method and its application to structural dynamic problems

A diagonal or lumped mass matrix is of great value for time-domain analysis of structural dynamic and wave propagation problems,as the computational efforts can be greatly reduced in the process of mass matrix inversion.In this study,the nodal quadrature method is employed to construct a lumped mass matrix for the Chebyshev spectral element method(CSEM).A Gauss-Lobatto type quadrature,based on Gauss-Lobatto-Chebyshev points with a weighting function of unity,is thus derived.With the aid of this quadrature,the CSEM can take advantage of explicit time-marching schemes and provide an efficient new tool for solving structural dynamic problems.Several types of lumped mass Chebyshev spectral elements are designed,including rod,beam and plate elements.The performance of the developed method is examined via some numerical examples of natural vibration and elastic wave propagation,accompanied by their comparison to that of traditional consistent-mass CSEM or the classical finite element method(FEM).Numerical results indicate that the proposed method displays comparable accuracy as its consistent-mass counterpart,and is more accurate than classical FEM.For the simulation of elastic wave propagation in structures induced by high-frequency loading,this method achieves satisfactory performance in accuracy and efficiency.

Chebyshev Pseudo-Spectral Method for Solving Fractional Advection-Dispersion Equation

Fractional differential equations have recently been applied in various areas of engineering, science, finance, applied mathematics, bio-engineering and others. However, many researchers remain unaware of this field. In this paper, an efficient numerical method for solving the fractional Advection-dispersion equation (ADE) is considered. The fractional derivative is described in the Caputo sense. The method is based on Chebyshev approximations. The properties of Chebyshev polynomials are used to reduce ADE to a system of ordinary differential equations, which are solved using the finite difference method (FDM). Moreover, the convergence analysis and an upper bound of the error for the derived formula are given. Numerical solutions of ADE are presented and the results are compared with the exact solution.

Chebyshev finite spectral method with extended moving grids

A Chebyshev finite spectral method on non-uniform meshes is proposed.An equidistribution scheme for two types of extended moving grids is used to generate grids. One type is designed to provide better resolution for the wave surface, and the other type is for highly variable gradients. The method has high-order accuracy because of the use of the Chebyshev polynomial as the basis function. The polynomial is used to interpolate the values between the two non-uniform meshes from a previous time step to the current time step. To attain high accuracy in the time discretization, the fourth-order Adams-Bashforth-M oulton predictor and corrector scheme is used. To avoid numerical oscillations caused by the dispersion term in the Korteweg-de Vries (KdV) equation,a numerical technique on non-uniform meshes is introduced. The proposed numerical scheme is validated by the applications to the Burgers equation (nonlinear convection-diffusion problems) and the KdV equation (single solitary and 2-solitary wave problems),where analytical solutions are available for comparisons. Numerical results agree very well with the corresponding analytical solutions in all cases.

Two-Level Block Decompositions for Solving Helmholtz Equation via Chebyshev Pseudo Spectral Method

In this paper, we consider solving the Helmholtz equation in the Cartesian domain , subject to homogeneous Dirichlet boundary condition, discretized with the Chebyshev pseudo-spectral method. The main purpose of this paper is to present the formulation of a two-level decomposition scheme for decoupling the linear system obtained from the discretization into independent subsystems. This scheme takes advantage of the homogeneity property of the physical problem along one direction to reduce a 2D problem to several 1D problems via a block diagonalization approach and the reflexivity property along the second direction to decompose each of the 1D problems to two independent subproblems using a reflexive decomposition, effectively doubling the number of subproblems. Based on the special structure of the coefficient matrix of the linear system derived from the discretization and a reflexivity property of the second-order Chebyshev differentiation matrix, we show that the decomposed submatrices exhibits a similar property, enabling the system to be decomposed using reflexive decompositions. Explicit forms of the decomposed submatrices are derived. The decomposition not only yields more efficient algorithm but introduces coarse-grain parallelism. Furthermore, it preserves all eigenvalues of the original matrix.

Numerical Solution of Klein/Sine-Gordon Equations by Spectral Method Coupled with Chebyshev Wavelets

The basic aim of this paper is to introduce and describe an efficient numerical scheme based on spectral approach coupled with Chebyshev wavelets for the approximate solutions of Klein-Gordon and Sine-Gordon equations. The main characteristic is that, it converts the given problem into a system of algebraic equations that can be solved easily with any of the usual methods. To show the accuracy and the efficiency of the method, several benchmark problems are implemented and the comparisons are given with other methods existing in the recent literature. The results of numerical tests confirm that the proposed method is superior to other existing ones and is highly accurate

THE ACCURACY COMPARISON BETWEEN CHEBYSHEV-τ METHOD AND CHEBYSHEV COLLOCATION METHOD

采用伪谱方法中的Ｃｈｅｂｙｓｈｅｖ－τ方法和配置点方法，分别研究了平面Ｐｏｉｓｅｕｉｌｅ流中扰动的非线性演化问题，并进行了比较．结果表明，在中性情况附近，配置点方法的精度要高于Ｃｈｅｂｙｓｈｅｖ－τ方法的精度．

Volterra型积分微分方程Chebyshev谱配置法求解

采用Chebyshev谱配置法求解Volterra型积分微分方程.首先将积分微分方程改写成等价的第二类Volterra积分方程组,再取Clenshaw-Curtis点为配置点,然后利用Clenshaw-Curtis求积法则离散方程中积分项得到配置方程组,最后给出在L∞范数空间下的误差分析,并用数值实例验证理论分析的结果.该方法既有谱精度,程序又易实现.

Spectral Method for Solving Time Dependent Flow of Upper-Convected Maxwell Fluid in Tube

The ti me dependent flow of upper-convected Maxwell fluid in a horizontal circular pip e is studied by spectral method. The time dependent problem is mathematically re duced to a partial differential equation of second order. By using spectral meth od the partial differential equation can be reduced to a system of ordinary diff erential equations for different terms of Chebyshev polynomials approximations. The ordinary differential equations are solved by Laplace transform and the eige nvalue method that leads to an analytical form of the solutions.

Direct spectral domain decomposition method for 2D incompressible Navier-Stokes equations

An efficient direct spectral domain decomposition method is developed coupled with Chebyshev spectral approximation for the solution of 2D, unsteady and incompressible Navier-Stokes equations in complex geometries. In this numerical approach,the spatial domains of interest are decomposed into several non-overlapping rectangular sub-domains. In each sub-domain, an improved projection scheme with second-order accuracy is used to deal with the coupling of velocity and pressure, and the Chebyshev collocation spectral method(CSM) is adopted to execute the spatial discretization. The influence matrix technique is employed to enforce the continuities of both variables and their normal derivatives between the adjacent sub-domains. The imposing of the Neumann boundary conditions to the Poisson equations of pressure and intermediate variable will result in the indeterminate solution. A new strategy of assuming the Dirichlet boundary conditions on interface and using the first-order normal derivatives as transmission conditions to keep the continuities of variables is proposed to overcome this trouble. Three test cases are used to verify the accuracy and efficiency, and the detailed comparison between the numerical results and the available solutions is done. The results indicate that the present method is efficiency, stability, and accuracy.

A Spectral Method for Convection-Diffusion Equations

In the practical problems such as nuclear waste pollution and seawater intrusion etc., many problems are reduced to solving the convection-diffusion equation, so the research of convection-diffusion equation is of great value. In this work, a spectral method is presented for solving one and two dimensional convection-diffusion equation with source term. The finite difference method is also used to solve the convection diffusion equation. The numerical experiments show that the spectral method is more efficient than other methods for solving the convection-diffusion equation.

Chebyshev谱方法研究非稳态Maxwell流体在轴向余弦振荡圆柱上的斜驻点流动

研究了非稳态Maxwell流体斜撞击轴向余弦振荡圆柱的斜驻点流动.首先,基于斜驻点流动特性,在柱面坐标系下求得关于压力的二阶常微分方程,对压强进行修正,建立了非稳态Maxwell流体在振荡圆柱上斜驻点流动的边界层模型.接着,合理的相似变换将模型转化,使用Chebyshev谱方法求得模型的数值解.结果表明,在贴近圆柱表面的流体随着圆柱体做周期性运动;圆柱的曲率越大越会使在同一时刻同一位置处的流体质点的速度越大;相反,非稳态参数及流体的记忆特性也会在更靠近圆柱壁面处阻碍流体流动.

A Spectral Method in Time for Initial-Value Problems

A time-spectral method for solution of initial value partial differential equations is outlined. Multivariate Chebyshev series are used to represent all temporal, spatial and physical parameter domains in this generalized weighted residual method (GWRM). The approximate solutions obtained are thus analytical, finite order multivariate polynomials. The method avoids time step limitations. To determine the spectral coefficients, a system of algebraic equations is solved iteratively. A root solver, with excellent global convergence properties, has been developed. Accuracy and efficiency are controlled by the number of included Chebyshev modes and by use of temporal and spatial subdomains. As examples of advanced application, stability problems within ideal and resistive magnetohydrodynamics (MHD) are solved. To introduce the method, solutions to a stiff ordinary differential equation are demonstrated and discussed. Subsequently, the GWRM is applied to the Burger and forced wave equations. Comparisons with the explicit Lax-Wendroff and implicit Crank-Nicolson finite difference methods show that the method is accurate and efficient. Thus the method shows potential for advanced initial value problems in fluid mechanics and MHD.

Chebyshev Spectral Element Analysis for Pore-Pressure of ERDs during Construction Period

Chebyshev spectral elements are applied to dissipation analysis of pore-pressure of roller compaction earth-rockfilled dams (ERD) during their construction. Nevertheless, the conventional finite element, for its excellent adaptability to complex geometrical configuration, is the most common way of spatial discretization for the pore-pressure solution of ERDs now [1]. The spectral element method, by means of the spectral isoparametric transformation, surmounts the disadvantages of disposing with complex geometry. According to the illustration of numerical examples, one can conclude that the spectral element methods have the following obvious advantages: 1) large spectral elements can be used in spectral element methods for the domains of homogeneous material;2) in the application of large spectral elements to spatial discretization, only a few leading terms of Chebyshev interpolation polynomial are taken to arrive at the solutions of better accuracy;3) spectral element methods have excellent convergence as well-known. Spectral method also is used to integrate the evolution equation in time to avoid the limitation of conditional stability of time-history

微通道中一类生物流体在高Zeta势下的电渗流及传热特性

在高壁面Zeta势下,研究滑移边界条件下满足牛顿流体模型的一类生物流体的电渗流动及传热特性,流体在外加电场、磁场和焦耳加热共同作用下流动.首先,在不使用Debye-Hückel线性近似条件时,利用切比雪夫谱方法给出非线性Poisson-Boltzmann方程和流函数满足的四阶微分方程及热能方程的数值解,将所得结果与利用Debye-Hückel线性近似所得结果进行比较,证明本文数值方法的有效性.其次,讨论电磁环境下壁面Zeta势、哈特曼数H、电渗参数m、滑移参数β对流动特性、泵送特性和捕获现象的影响,并探究焦耳加热参数γ和布林克曼数Br等参数对传热特性的影响.结果表明,壁面Zeta势、电渗参数m、滑移参数β的增大对流体速度有促进作用,而哈特曼数H的增大会抵抗流体流动.研究进一步表明,焦耳加热参数γ和布林克曼数Br的增大会导致温度升高.

极坐标与圆柱坐标下Fourier-Chebyshev配置点谱方法泊松方程求解器

采用矩阵相乘的Fourier-Chebyshev配置点谱方法求解极坐标与圆柱坐标系下的泊松方程.通常,在极坐标与圆柱坐标系下运用谱方法求解泊松方程会产生奇点问题.为了避免这个问题,分别采用两种方法开发了泊松方程求解器.一种方法是采用Gauss-Radau配置点,从而排除中心点r=0;另一种方法是采用区域转换将半径方向计算域[0,1]转换成[-1,1],采用Gauss-Lobatto配置点,当节点数取奇数时同样避开了中心点r=0.这两种方法均避免了中心处的奇点,且不需构造额外的极条件.针对二维、三维的不同算例进行了比较和验证计算.计算结果证明两个求解器都具有直接、快速、高精度的特性.

Chebyshev超谱粘性法在推进剂供应管路非定常流动分析中的应用

基于一维管道瞬变流理论和数值谱方法,给出了求解推进剂供应系统管路内液体非定常流动控制方程的Chebyshev超谱粘性法。与常规谱方法相比,该方法有效地克服了由于解的间断或大梯度变化而发生的非物理振荡现象,而且在一定程度上加快了收敛速度,提高了计算效率。以一段两端分别连接贮箱和阀门的等截面圆直管为例,利用该方法对阀门关闭后管道内水击现象进行了计算,给出了相应的水击压力仿真结果,并分别与采用特征线方法和传统谱方法求得的结果进行了对比分析,验证了方法的正确性和优越性。