广义KdV方程Fourier谱逼近的最优误差估计

A Fourier spectral method for the generalized Korteweg-de Vries equation with periodic boundary conditions is analyzed, and a corresponding optimal error estimate in L^2-norm is obtained. It improves the result presented by Maday and Quarteroni. A modified Fourier pseudospectral method is also presented, with the same convergence properties as the Fourier spectral method.     

SPECTRAL METHOD IN TIME FOR KdV EQUATIONS

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Mixed spectral method for exterior problems of Navier-Stokes equations by using generalized Laguerre functions

In this paper, we investigate the mixed spectral method using generalized Laguerre functions for exterior problems of fourth order partial differential equations. A mixed spectral scheme is provided for the stream function form of the Navier-Stokes equations outside a disc. Numerical results demonstrate the spectral accuracy in space.     

Analysis of chebyshev pseudospectral method for multi-dimensional generalized srlw equations

IntroductionAsymmetricregularizedlongwaveequation (SRLWE) 2 x2 -1 u t = x ρ+ 12 u2 ,　 ρ t+ u x=0 (1 )hasbeeninvestigatedinRef.[1 ] .Thesystem (1 )ofequationsisshowntodescribeweaklynonlinearion_acousticwaveandspace_chargewaves.Thehuperbolicsecantsquaredsolitarywaves ,thefourconservationlaws,andsomenumericalresultshavebeenobtainedinRef.[1 ] .Obviously ,eliminatingρin (1 ) ,weobtainaclassofregularizedlongwaveequationutt-uxx+ 12 u2xt-uxxtt =0 . (2 )TheSRLWequationisexplicitlysymmetric…     

Chebyshev collocation method and multidomain decomposition for the incompressible Navier-Stokes equations

The two-dimensional incompressible Navier-Stokes equations in primitive variables have been solved by a pseudospectral Chebyshev method using a semi-implicit fractional step scheme. The latter has been adapted to the particular features of spectral collocation methods to develop the monodomain algorithm. In particular, pressure and velocity collocated on the same nodes are sought in a polynomial space of the same order; the cascade of scalar elliptic problems arising after the spatial collocation is solved using finite difference preconditioning. With the present procedure spurious pressure modes do not pollute the pressure field. As a natural development of the present work a multidomain extent was devised and tested. The original domain is divided into a union of patching sub-rectangles. Each scalar problem obtained after spatial collocation is solved by iterating by subdomains. For steady problems a C¹ solution is recovered at the interfaces upon convergence, ensuring a spectrally accurate solution. A number of test cases have been solved to validate the algorithm in both its single-block and multidomain configurations. The preliminary results achieved indicate that collocation methods in multidomain configurations might become a viable alternative to the spectral element technique for accurate flow prediction.     

A new multistage spectral relaxation method for solving chaotic initial value systems

In this paper, we present a new pseudospectral method application for solving nonlinear initial value problems (IVPs) with chaotic properties. The proposed method, called the multistage spectral relaxation method (MSRM) is based on a novel technique of extending Gauss–Siedel type relaxation ideas to systems of nonlinear differential equations and using the Chebyshev pseudo-spectral methods to solve the resulting system on a sequence of multiple intervals. In this new application, the MSRM is used to solve famous chaotic systems such as the such as Lorenz, Chen, Liu, Rikitake, Rössler, Genesio–Tesi, and Arneodo–Coullet chaotic systems. The accuracy and validity of the proposed method is tested against Runge–Kutta and Adams–Bashforth–Moulton based methods. The numerical results indicate that the MSRM is an accurate, efficient, and reliable method for solving very complex IVPs with chaotic behavior.     

On solitons in media modelled by the hierarchical KdV equation

In the present paper, formation of solitons in microstructured continuum, modelled by a hierarchical Korteweg–de Vries equation, is studied. The model equation is integrated numerically making use of the discrete Fourier transform-based pseudospectral method under different initial conditions. Main attention is paid to the formation of hidden solitons and applicability of the discrete spectral analysis.     

Multidomain pseudospectral methods for nonlinear convection-diffusion equations

Multidomain pseudospectral approximations to nonlinear convection-diffusion equations are considered. The schemes are formulated with the Legendre-Galerkin method, but the nonlinear term is collocated at the Legendre/Chebyshev-Gauss-Lobatto points inside each subinterval. Appropriate base functions are introduced so that the matrix of the system is sparse, and the method can be implemented efficiently and in parallel. The stability and the optimal rate of convergence of the methods are proved. Numerical results are given for both the single domain and the multidomain methods to make a comparison.     

Comparison of three spectral methods for the Benjamin-Ono equation: Fourier pseudospectral, rational Christov functions and Gaussian radial basis functions

The Benjamin-Ono equation is especially challenging for numerical methods because (i) it contains the Hilbert transform, a nonlocal integral operator, and (ii) its solitary waves decay only as O(1/|x|²). We compare three different spectral methods for solving this one-space-dimensional equation. The Fourier pseudospectral method is very fast through use of the Fast Fourier Transform (FFT), but requires domain truncation: replacement of the infinite interval by a large but finite domain. Such truncation is unnecessary for a rational basis, but it is simple to evaluate the Hilbert Transform only when the usual rational Chebyshev functions TB_n(x) are replaced by their cousins, the Christov functions; the FFT still applies. Radial basis functions (RBFs) are slow for a given number of grid points N because of the absence of a summation algorithm as fast as the FFT; because RBFs are meshless, however, very flexible grid adaptation is possible.     

Comparison of an Adaptive Wavelet Method and Nonlinearly Filtered Pseudospectral Methods for Two-Dimensional Turbulence

An adaptive wavelet method for solving the two-dimensional Navier–Stokes equations is compared with nonlinear Fourier filtering and nonlinear wavelet filtering of the pseudospectral method at each time step. The methods are each applied to a highly nonlinear flow typical of two-dimensional turbulence, the merger of two positive vortices pushed together by a weaker negative vortex, and the results are compared with a reference classical pseudospectral method. Nonlinear Fourier filtering uses 1.7 times fewer active modes than the reference simulation at the time of merger (when the flow is most complicated) and retains the overall dynamics and structure of the flow. However, it induces spurious oscillations in the background. Nonlinear wavelet filtering simulation uses 9.2 times fewer modes than the reference simulation at the time of merger, and reduces the errors in the solution. The adaptive wavelet simulation replicates precisely the dynamics and spatial structure of the reference simulation while retaining the high compression rate of the nonlinear wavelet filtering simulation. In addition we observe that the number of active wavelet modes remains quasi-constant during the whole merging process, independent of the strength of the vorticity gradients. On the contrary, the number of active Fourier modes is multiplied by 5 when the vorticity gradients are strongest. The increased accuracy of the adaptive wavelet simulation is due to the security zone added around the active coefficients and to the compression of the nonlinear term of the Navier–Stokes equations in the wavelet basis. These results suggest that nonlinear Fourier filtering of a classical pseudospectral method cannot produce significant improvement, but that the adaptive wavelet method combines a consistently high compression rate with high accuracy. Received 22 April 1997 and accepted 11 August 1997     

自由漂浮空间机器人路径优化的Legendre伪谱法

基于Legendre 伪谱法研究自由漂浮空间机器人非完整路径规划的最优控制问题. 自由漂浮是空间机器人执行任务常用的工作模式,其路径优化是空间机器人完成复杂空间任务的基础. 由于空间机器人不具有固定基座,机械臂和载体之间存在非完整约束,使得自由漂浮空间机器人路径规划完全不同于地面机器人而变得具有挑战性. 本文提出自由漂浮空间机器人路径规划的最优控制伪谱方法. 首先,利用多体动力学理论建立自由漂浮空间机器人动力学模型,给定系统的初始和目标位形,选取机械臂关节耗散能最小为性能指标,并考虑实际控制输入受限,建立其路径规划的Bolza 问题. 然后,应用Legendre 伪谱法,将状态和控制变量在Legendre-Gauss-Lobatto (LGL) 点上离散,并构造Lagrange 插值多项式逼近系统状态和控制变量,将连续路径优化问题离散化为非线性规划问题求解. 最后通过数值仿真表明,应用Legendre 伪谱法求解自由漂浮空间机器人非完整路径规划问题,得到的机械臂和载体最优运动轨迹,较好地满足各种约束条件,且计算精度高、速度快,并具有良好的实时性.      

Spectral methods for the viscoelastic time-dependent flow equations with applications to Taylor–Couette flow

The time evolution of finite amplitude axisymmetric perturbations (Taylor cells) to the purely azimuthal, viscoelastic, cylindrical Couette flow was numerically simulated. Two time integration numerical methods were developed, both based on a pseudospectral spatial approximation of the variables, efficiently implemented using fast Poisson solvers and optimal filtering routines. The first method, applicable for finite Re numbers, is based on a time-splitting integration with the divergence-free condition enforced through an influence matrix technique. The second one, is based on a semi-implicit time integration of the constitutive equation with both the continuity and the momentum equations enforced as constraints. Stability results for an upper convected Maxwell fluid were obtained for the supercritical bifurcations, either steady or time-periodic, developed after the onset of instabilities in the primary flow. At small elasticity values, ? ≡ De/Re, the time integration of finite amplitude disturbances confirms the stability of the single branch of steady Taylor cells. At intermediate ? values the rotating wave family of time-periodic solutions developed at the onset of instability is stable, whereas the standing wave is found to be unstable. At high ? values, and in particular for the limit of creeping flow (? = ∞), the present study shows that the rotating wave family is unstable and the standing (radial) wave is stable, in agreement with previous finite-element investigations. It is thus shown that spectral techniques provide a robust and computationally efficient method for the simulation of complex, non-linear, time-dependent viscoelastic flows.     

A high-accuracy temporal–spatial pseudospectral method for time-periodic unsteady fluid flow and heat transfer problems

A temporal–spatial pseudospectral (TSP) method is proposed for the high-accuracy solutions of time-periodic unsteady fluid flow and heat transfer problems. In this method, both the spatial and temporal derivative terms in the governing equations are computed by pseudospectral method. The spatial derivatives are computed through Chebyshev and Lagrange polynomials while the time derivatives are computed by Fourier series. The TSP method is capable of directly finding out the periodic state solutions without the necessity to resolve the initial transient state solutions, hence holds high computational efficiency and high numerical accuracy properties for the time-periodic problems. This method is validated by three 2D benchmark problems: the time-periodic incompressible flow with exact solutions; the natural convection in enclosure with time-periodic temperature on one sidewall, and on both sidewalls. The TSP results fit well the exact solutions or the benchmark solutions and the TSP accuracy is much higher than the time marching spatial pseudospectral accuracy. Some time-dependent fluid flow and heat transfer characteristic parameters are analysed. The proposed TSP method could be further extended to more complex time-periodic unsteady fluid flow and heat transfer problems where high-accuracy results are required.     

Large‐eddy simulation with complex 2‐D geometries using a parallel finite‐element/spectral algorithm

A parallel stabilized finite‐element/spectral formulation is presented for incompressible large‐eddy simulation with complex 2‐D geometries. A unique discretization scheme is developed consisting of a streamline‐upwind Petrov–Galerkin/Pressure‐Stabilized Petrov–Galerkin (SUPG/PSPG) finite‐element discretization in the 2‐D plane with a collocated spectral/pseudospectral discretization in the out‐of‐plane direction. This formulation provides an efficient approach for solving 3‐D flows over arbitrary 2‐D geometries. Utilizing this discretization and through explicit temporal treatment of the non‐linear terms, the system of equations for each Fourier mode is decoupled within each time step. A novel parallelization approach is then taken, where the computational work is partitioned in Fourier space. A validation of the algorithm is presented via comparison of results for flow past a circular cylinder with published values for Re=195, 300, and 3900. Copyright © 2003 John Wiley & Sons, Ltd.     

Thermal and tensile loading effects on size-dependent vibration response of traveling nanobeam by wavelet-based spectral element modeling

This paper presents a novel approach for size-dependent vibration response of a nanostructure under tensile and thermal loads, traveling in its axial direction at a constant velocity. The traveling nanostructure is modeled as an Euler–Bernoulli nanobeam based on modified couple stress theory. Wavelet-based spectral element model (WSEM) is performed for analyzing vibration of the system. Imposing WSEM reduces the governing partial differential equation of the system to a set of ordinary differential equations. The roles of nanobeam velocity, tensile and thermal loads on vibration and wave characteristics, and divergence/flutter instability are scrutinized by WSEM. The validity and accuracy of resulting responses are inspected by comparing with numeric values obtained from spectral element and finite element methods, and whenever possible, with those available in the literature.     

On the application of pseudospectral FFT techniques to non-periodic problems

The reduction-to-periodicity method using the pseudospectral fast Fourier transform (FFT) technique is applied to the solution of non-periodic problems, including the two-dimensional incompressible Navier–Stokes equations. The accuracy of the method is explored by calculating the derivatives of given functions, one- and two-dimensional convective-diffusive problems, and by comparing the relative errors due to the FFT method with a second-order finite difference (FD) method. Finally, the two-dimensional Navier–Stokes equations are solved by a fractional step procedure using both the FFT and the FD methods for the driven cavity flow and the backward-facing step problems. Comparisons of these solutions provide a realistic assessment of the FFT method.     

大跨度桥梁空间脉动风场的计算机模拟

针对现有Deodatis方法模拟大跨度桥梁空间脉动风场中存在的计算量问题,通过对谱分解矩阵引入插值近似,减少谱分解的次数,从而提高该谐波合成法的计算效率,并节省内存花费,实现了对三维空间脉动风场的有效模拟。改进方法模拟的脉动风速样本仍保持各态历经性,且逐渐收敛到目标功率谱。用改进的Deodatis方法模拟了润扬长江悬索桥桥面主梁上作用的纵向脉动风速。结果表明,该改进措施对Deodatis方法的应用效果非常明显,改进的Deodatis方法模拟脉动风速样本的相关函数与目标相关函数均吻合良好。尽管改进后的Deo-datis方法对谱分解矩阵采用了插值近似,但模拟的随机风速样本仍具有很好的精度。     

A pseudospectral approach applicable for time integration of linearized N-S operator that removes pole singularity and physically spurious eigenmodes

This paper addresses a modified singularity removal technique for the eigenvalue or optimal mode problems in pipe flow using a pseudospectral method. The current approach results in the linear stability operator to be devoid of any unstable physically spurious modes, and thus, it provides higher numerical stability during time-based integration. The correctness of the numerical operator is established by calculating the known eigenvalues of pipe Poiseuille flow. Subsequently, the optimal modes are determined with Farrell's approach and compared with the existing literature. The usefulness of this approach is further demonstrated in the time-based numerical integration of the linearized Navier-Stokes operator for the adjoint method–based optimal mode determination. The numerical scheme is implemented with the radial velocity-radial vorticity formulation. Even number of Chebyshev-Lobatto grid points are distributed over the domain r∈[−1,1] omitting the centerline, which also efficiently provides higher resolution near the wall boundary. The boundary conditions are imposed with homogeneous wall boundary conditions, whereas the analytic nature of a proper set of base functions enforces correct centerline conditions. The resulting redundancy introduced in the process is eliminated with the proper usage of parity.     

Invariants of a Remarkable Family of Nonlinear Equations

In classical literature, invariants of families of differentialequations were considered for linear equations only, e.g. the renownedLaplace invariants for linear hyperbolic partial differential equationsand invariants of linear ordinary differential equations with variablecoefficients. The restriction to linear equations was essential inpioneering works of Cockle, Laguerre, Halphen, andForsyth for tackling the problem of invariants of differentialequations. Lie regretted that these authors did not use advantagesprovided by his theory of infinite continuous groups, but he himself didnot undertake further developments in this direction.Recently, the present author considered the possibility hinted byLie's remark and introduced the infinitesimal technique in thetheory of invariants of families of differential equations thatwas lacking in old methods. In consequence, a simple unifiedapproach was developed for calculation of invariants of algebraicand differential equations independent on the assumption oflinearity of equations. It was employed recently for calculationof Laplace type invariants for parabolic equations. Here, themethod is applied to calculation of invariants for the family ofnonlinear equations appearing in the problem on linearization ofnonlinear ordinary differential equations.