A Legendre–Galerkin Chebyshev collocation method for the Burgers equation with a random perturbation on boundary condition

In the paper, we apply the generalized polynomial chaos expansion and spectral methods to the Burgers equation with a random perturbation on its left boundary condition. Firstly, the stochastic Galerkin method combined with the Legendre–Galerkin Chebyshev collocation scheme is adopted, which means that the original equation is transformed to the deterministic nonlinear equations by the stochastic Galerkin method and the Legendre–Galerkin Chebyshev collocation scheme is used to deal with the resulting nonlinear equations. Secondly, the stochastic Legendre–Galerkin Chebyshev collocation scheme is developed for solving the stochastic Burgers equation; that is, the stochastic Legendre–Galerkin method is used to discrete the random variable meanwhile the nonlinear term is interpolated through the Chebyshev–Gauss points. Then a set of deterministic linear equations can be obtained, which is in contrast to the other existing methods for the stochastic Burgers equation. The mean square convergence of the former method is analyzed. Numerical experiments are performed to show the effectiveness of our two methods. Both methods provide alternative approaches to deal with the stochastic differential equations with nonlinear terms.     

A-Posteriori Error Estimation for the Legendre Spectral Galerkin Method in One-Dimension

<正>In this paper,a-posteriori error estimators are proposed for the Legendre spectral Galerkin method for two-point boundary value problems.The key idea is to postprocess the Galerkin approximation,and the analysis shows that the postprocess improves the order of convergence.Consequently,we obtain asymptotically exact aposteriori error estimators based on the postprocessing results.Numerical examples are included to illustrate the theoretical analysis.     

Galerkin方法的一种简单后处理过程

基于近似惯性流形思想,以流函数形式定常Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程为例,给出了一种简单的后处理Ｇａｌｅｒｋｉｎ方法。其主要思想是利用近似惯性流形概念和对真解的一种新的分解,构造高低频分量间的近似作用规律。文中证明了这种简单的后处理Ｇａｌｅｒｋｉｎ方法可以较小的代价获得较经典Ｇａｌｅｒｋｉｎ方法高得多的精度。     

A Method for Solving Fredholm Integral Equations of the First Kind Based on Chebyshev Wavelets

In this paper, we suggest a method for solving Fredholm integral equation of the first kind based on wavelet basis. The continuous Legendre and Chebyshev wavelets of the first, second, third and fourth kind on [0,1] are used and are utilized as a basis in Galerkin method to approximate the solution of integral equations. Then, in some examples the mentioned wavelets are compared with each other.     

Stochastic Approaches to Uncertainty Quantification in CFD Simulations

This paper discusses two stochastic approaches to computing the propagation of uncertainty in numerical simulations: polynomial chaos and stochastic collocation. Chebyshev polynomials are used in both cases for the conventional, deterministic portion of the discretization in physical space. For the stochastic parameters, polynomial chaos utilizes a Galerkin approximation based upon expansions in Hermite polynomials, whereas stochastic collocation rests upon a novel transformation between the stochastic space and an artificial space. In our present implementation of stochastic collocation, Legendre interpolating polynomials are employed. These methods are discussed in the specific context of a quasi-one-dimensional nozzle flow with uncertainty in inlet conditions and nozzle shape. It is shown that both stochastic approaches efficiently handle uncertainty propagation. Furthermore, these approaches enable computation of statistical moments of arbitrary order in a much more effective way than other usual techniques such as the Monte Carlo simulation or perturbation methods. The numerical results indicate that the stochastic collocation method is substantially more efficient than the full Galerkin, polynomial chaos method. Moreover, the stochastic collocation method extends readily to highly nonlinear equations. An important application is to the stochastic Riemann problem, which is of particular interest for spectral discontinuous Galerkin methods.     

Efficient methods using high accuracy approximate inertial manifolds

Summary. We extend the idea of the post-processing Galerkin method, in the context of dissipative evolution equations, to the nonlinear Galerkin, the filtered Galerkin, and the filtered nonlinear Galerkin methods. In general, the post-processing algorithm takes advantage of the fact that the error committed in the lower modes of the nonlinear Galerkin method (and Galerkin method), for approximating smooth, bounded solutions, is much smaller than the total error of the method. In each case, an improvement in accuracy is obtained by post-processing these more accurate lower modes with an appropriately chosen, highly accurate, approximate inertial manifold (AIM). We present numerical experiments that support the theoretical improvements in accuracy. Both the theory and computations are presented in the framework of a two dimensional reaction-diffusion system with polynomial nonlinearity. However, the algorithm is very general and can be implemented for other dissipative evolution systems. The computations clearly show the post-processed filtered Galerkin method to be the most efficient method. Received September 10, 1998 / Revised version received April 26, 1999 / Published online July 12, 2000     

Stochastic approaches to uncertainty quantification in CFD simulations

This paper discusses two stochastic approaches to computing the propagation of uncertainty in numerical simulations: polynomial chaos and stochastic collocation. Chebyshev polynomials are used in both cases for the conventional, deterministic portion of the discretization in physical space. For the stochastic parameters, polynomial chaos utilizes a Galerkin approximation based upon expansions in Hermite polynomials, whereas stochastic collocation rests upon a novel transformation between the stochastic space and an artificial space. In our present implementation of stochastic collocation, Legendre interpolating polynomials are employed. These methods are discussed in the specific context of a quasi-one-dimensional nozzle flow with uncertainty in inlet conditions and nozzle shape. It is shown that both stochastic approaches efficiently handle uncertainty propagation. Furthermore, these approaches enable computation of statistical moments of arbitrary order in a much more effective way than other usual techniques such as the Monte Carlo simulation or perturbation methods. The numerical results indicate that the stochastic collocation method is substantially more efficient than the full Galerkin, polynomial chaos method. Moreover, the stochastic collocation method extends readily to highly nonlinear equations. An important application is to the stochastic Riemann problem, which is of particular interest for spectral discontinuous Galerkin methods.     

The Chebyshev–Shamanskii Method for Solving Systems of Nonlinear Equations

We present a method, based on the Chebyshev third-order algorithm and accelerated by a Shamanskii-like process, for solving nonlinear systems of equations. We show that this new method has a quintic convergence order. We will also focus on efficiency of high-order methods and more precisely on our new Chebyshev–Shamanskii method. We also identify the optimal use of the same Jacobian in the Shamanskii process applied to the Chebyshev method. Some numerical illustrations will confirm our theoretical analysis.     

A rational spectral method for the KdV equation on the half line

A Petrov–Galerkin method using orthogonal rational functions is proposed for the Korteweg–de Vries (KdV) equation on the half line with initial-boundary values. The nonlinear term and the right-hand side term are treated by Chebyshev rational interpolation explicitly, and the linear terms are computed with the Galerkin method implicitly. Such an approach is applicable using fast algorithms. Numerical results are presented for problems with both exponentially and algebraically decaying solutions, respectively, highlighting the performance of the proposed method.     

A new operational matrix of fractional order integration for the Chebyshev wavelets and its application for nonlinear fractional Van der Pol oscillator equation

In this paper, an efficient and accurate computational method based on the Chebyshev wavelets (CWs) together with spectral Galerkin method is proposed for solving a class of nonlinear multi-order fractional differential equations (NMFDEs). To do this, a new operational matrix of fractional order integration in the Riemann–Liouville sense for the CWs is derived. Hat functions (HFs) and the collocation method are employed to derive a general procedure for forming this matrix. By using the CWs and their operational matrix of fractional order integration and Galerkin method, the problems under consideration are transformed into corresponding nonlinear systems of algebraic equations, which can be simply solved. Moreover, a new technique for computing nonlinear terms in such problems is presented. Convergence of the CWs expansion in one dimension is investigated. Furthermore, the efficiency and accuracy of the proposed method are shown on some concrete examples. The obtained results reveal that the proposed method is very accurate and efficient. As a useful application, the proposed method is applied to obtain an approximate solution for the fractional order Van der Pol oscillator (VPO) equation.     

加罚N-S方程的有限元非线性Galerkin方法

非线性Galerkin方法是对耗散型非线性发展方程的一种数值解法,其空间变量不象一般Galerkin方法那样在线性空间上离散,而是在非线性流形上离散,所得逼近解在时间变量增大时可以更快地逼近其精确解.精细的理论分析可见[1],[2]等,在有限元逼近基础上将此方法应用到Navier-Stokes方程上的工作可参见[3],[4],这些文章主要针对速度与压力同时求解的混合元情形做了讨论.本文在[4]的基础上对加罚Navier-Stokes方程的一种非线性Galerkin方法的半离散和全离散有限元逼近格式分别进行了误差估     

Second Kind Chebyshev Wavelet Galerkin Method for Stochastic Itô-Volterra Integral Equations

In this paper, an efficient wavelet Galerkin method based on the stochastic operational matrix of second kind Chebyshev wavelet is proposed for solving stochastic Itô-Volterra integral equations. Convergence and error analysis of the presented wavelets method are investigated. The numerical results are compared with exact solution and those of other existing methods.     

An efficient pseudo‐spectral Legendre–Galerkin method for solving a nonlinear partial integro‐differential equation arising in population dynamics

The pseudo‐spectral Legendre–Galerkin method (PS‐LGM) is applied to solve a nonlinear partial integro‐differential equation arising in population dynamics. This equation is a competition model in which similar individuals are competing for the same resources. It is a kind of reaction–diffusion equation with integral term corresponding to nonlocal consumption of resources. The proposed method is based on the Legendre–Galerkin formulation for the linear terms and interpolation operator at the Chebyshev–Gauss–Lobatto (CGL) points for the nonlinear terms. Also, the integral term, which is a kind of convolution, is directly computed by a fast and accurate method based on CGL interpolation operator, and thus, the use of any quadrature formula in its computation is avoided. The main difference of the PS‐LGM presented in the current paper with the classic LGM is in treating the nonlinear terms and imposing boundary conditions. Indeed, in the PS‐LGM, the nonlinear terms are efficiently handled using the CGL points, and also the boundary conditions are imposed strongly as collocation methods. Combination of the PS‐LGM with a semi‐implicit time integration method such as second‐order backward differentiation formula and Adams‐Bashforth method leads to reducing the complexity of computations and obtaining a linear algebraic system of equations with banded coefficient matrix. The desired equation is considered on one and two‐dimensional spatial domains. Efficiency, accuracy, and convergence of the proposed method are demonstrated numerically in both cases. Copyright © 2012 John Wiley & Sons, Ltd.     

On the convergence of the collocation method for nonlinear boundary integral equations

Recently, Galerkin and collocation methods have been analysed for some nonlinear boundary integral equations. For the collocation method it has been assumed that the nonlinearity is asymptotically linear. In this paper we remove this restriction. We shall prove the convergence of the collocation method for nonlinear boundary integral equations, when the nonlinearity has a polynomial growth condition. In addition to this the optimal order error estimates follow in L^q(Γ)-norm.     

Chebyshev-Secant-type Methods for Non-differentiable Operators

In this paper we give a semilocal convergence theorem for a family of iterative methods for solving nonlinear equations defined between two Banach spaces. This family is obtained as a combination of the well known Secant method and Chebyshev method. We give a very general convergence result that allow the application of these methods to non-differentiable problems.     

ERROR ANALYSIS OF NONLINEAR GALERKIN METHODS FOR KURAMOTO-SIVASHINSKY EQUATIONS

Nonlinear Galerkin methods are new schemes for integrating dissipative systems:In the present paper, we obtain the estimates to the rate of convergence of such methods for Kuramoto-Sivashinsky equations. In particular, by an illustrative example, we show that nonlinear Galerkin methods converge faster than the usual Galerkin method.     

Chebyshev‐Legendre spectral method for solving the two‐dimensional vorticity equations with homogeneous Dirichlet conditions

The Chebyshev‐Legendre spectral method for the two‐dimensional vorticity equations is considered. The Legendre Galerkin Chebyshev collocation method is used with the Chebyshev‐Gauss collocation points. The numerical analysis results under the L²‐norm for the Chebyshev‐Legendre method of one‐dimensional case are generalized into that of the two‐dimensional case. The stability and optimal order convergence of the method are proved. Numerical results are given. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Sparse regression Chebyshev polynomial interval method for nonlinear dynamic systems under uncertainty

This paper proposes a new higher-efficiency interval method for the response bound estimation of nonlinear dynamic systems, whose uncertain parameters are bounded. This proposed method uses sparse regression and Chebyshev polynomials to help the interval analysis applied on the estimation. It is also a non-intrusive method which needs much fewer evaluations of original nonlinear dynamic systems than the other Chebyshev polynomials based interval methods. By using the proposed method, the response bound estimation of nonlinear dynamic systems can be performed more easily, even if the numerical simulation in nonlinear dynamic systems is costly or the number of uncertain parameters is higher than usual. In our approach, the sparse regression method “elastic net” is adopted to improve the sampling efficiency, but with sufficient accuracy. It alleviates the sample size required in coefficient calculation of the Chebyshev inclusion function in the sampling based methods. Moreover, some mature technologies are adopted to further reduce the sample size and to guarantee the accuracy of the estimation. So that the number of sampling, which solves the certain ordinary differential equations (ODEs), can be reduced significantly in the Chebyshev interval method. Three numerical examples are presented to illustrate the efficiency of proposed interval method. In particular, the last two examples are high dimension uncertain problems, which can further exhibit the ability to reduce the computational cost.     

Application of the Galerkin method to the formulation of a three-dimensional nonlinear hydrodynamic numerical sea model

A mathematical formulation is presented for solving the three-dimensional nonlinear hydrodynamic equations, using the Galerkin method with an arbitrary set of basis functions.An explicit time splitting method is used to integrate these equations through time. The time splitting method is formulated in such a way that the advective terms, which are computationally expensive to evaluate, are integrated with a longer time step than the linear terms. The length of the time step used to integrate the linear terms is determined by the propagation speed of the gravity waves. The paper demonstrates that using this time splitting method an accurate and computationally economic solution of the full three-dimensional equations is possible.Numerical results are presented for the nonlinear seiche motion in a one-dimensional basin, and for the three-dimensional wind induced flow in a closed rectangular basin, using basis sets of cosine functions, Chebyshev polynomials and Gram-Schmidt orthogonalized polynomials.     

On the numerical treatment of the eigenparameter dependent boundary conditions

In this paper, we consider the numerical treatment of singular eigenvalue problems supplied with eigenparameter dependent boundary conditions using spectral methods. On the one hand, such boundary conditions hinder the construction of test and trial space functions which could incorporate them and thus providing well-conditioned Galerkin discretization matrices. On the other hand, they can generate surprising behavior of the eigenvectors hardly detected by analytic methods. These singular problems are often indirectly approximated by regular ones. We argue that spectral collocation as well as tau method offer remedies for the first two issues and provide direct and efficient treatment to such problems. On a finite domain, we consider the so-called Petterson-König’s rod eigenvalue problem and on the half line, we take into account the Charney’s baroclinic stability problem and the Fourier eigenvalue problem. One boundary condition in these problems depends on the eigenparameter and additionally, this also could depend on some physical parameters. The Chebyshev collocation based on both, square and rectangular differentiation and a Chebyshev tau method are used to discretize the first problem. All these schemes cast the problems into singular algebraic generalized eigenvalue ones which are solved by the QZ and/or Arnoldi algorithms as well as by some target oriented Jacobi-Davidson methods. Thus, the spurious eigenvalues are completely eliminated. The accuracy of square Chebyshev collocation is roughly estimated and its order of approximation with respect to the eigenvalue of interest is determined. For the problems defined on the half line, we make use of the Laguerre-Gauss-Radau collocation. The method proved to be reliable, accurate, and stable with respect to the order of approximation and the scaling parameter.