关于二维Navier-Stokes方程非线性Galerkin校正的注记

本文在对二维Navier-Stokes方程及其近似惯性流形领域估计的基础上，讨论了非线性Galerkin方法校正量的一种可能的最优选取。     

Inertial manifolds of parabolic differential equations under high-order discretizations

This paper deals with the long-time behaviour of numerical discretizationsof nonlinear parabolic differential equations. For various equationsof mathematical physics, the dynamics are governed by a finite-dimensionalinertial manifold, which attracts solutions at an exponentialrate. We show that Runge-Kutta time and spectral Galerkin spacediscretizations possess inertial manifolds which approximatethe inertial manifold of the continuous problem with the orderof finite-time approximations of smooth solutions. We thus obtainestimates for the distance between the inertial manifolds ofthe partial differential equation and its semi- and full discretizationswhich show the high order of the time discretization and exponentiallyfast convergence of the space discretization. These resultsare obtained by using time analyticity and Gevrey regularityof solutions of the differential equation. As an applicationof the theory, the complex Ginzburg-Landau equation is considered.     

An applicatin of approximate inertial manifolds to a weakly damped nonlinear schrödinger equation

Nonlinear Galerkin methods (NGMs) based on pproximate inertial manifolds are applied to a weakly dissipative nonlinear Schrödinger equation. The purpose is to capture critical and chaotic behavior with as few modes as possible. Density functions are used on both the energy and instantaneous Lyapunov exponents to determine convergence of a chaotic attractor as the number of modes is increased. The computations presented here indicate a substantial reduction in the number of modes needed for the NGMs, compared to that needed for the traditional Galerkin method.     

广义Burgers方程的Legendre-Galerkin Chebyshev-配置方法

本文对广义Burgers方程的Neumann和Robin型边值问题构造了LegendreGalerkinChebyshev-配置方法.Legendre-GalerkinChebyshev-配置方法整体上按LegendreGalerkin方法形成,但对非线性项采用在Chebyshev-Gauss-Lobatto点上的配置法处理.文中给出了方法的稳定性和收敛性分析,获得了按H1-模的最佳误差估计.数值实验证实了方法的有效性.     

A postprocessed Galerkin method with Chebyshev or Legendre polynomials

Summary. We present an approximate-inertial-manifold-based postprocess to enhance Chebyshev or Legendre spectral Galerkin methods. We prove that the postprocess improves the order of convergence of the Galerkin solution, yielding the same accuracy as the nonlinear Galerkin method. Numerical experiments show that the new method is computationally more efficient than Galerkin and nonlinear Galerkin methods. New approximation results for Chebyshev polynomials are presented. Received January 5, 1998 / Revised version received September 7, 1999 / Published online June 8, 2000     

Approximate inertial manifolds of Burgers equation approached by nonlinear Galerkin’s procedure and its application

The approximate inertial manifolds (AIMs) of Burgers equation is approached by nonlinear Galerkin methods, and it can be used to capture and study the shock wave numerically in a reduced system with low dimension. Following inertial manifolds, the asymptotic behavior of Burgers equation, an infinite dimensional dissipative dynamic systems, will evolve to a compact set known as a global attractor, which is finite-dimensional, and the nonlinear phenomena are included and captured in such global attractor. In the application, nonlinear Galerkin methods is introduced to approach such inertial manifolds. By this method, the solution of the original system is projected onto the complete space spanned by the eigenfunctions or the modes of the linear operator of Burgers equation, and nonlinear Galerkin method splits the infinite-dimensional phase space into two complementary subspaces: a finite-dimensional one and its infinite-dimensional complement. Then, the post-processed Galerkin’s procedure is used to approximate the solution of the reduced system, with the introduction of the interaction between lower and higher modes. Additionally, some numerical examples are presented to make a comparison between the traditional Galerkin method and nonlinear Galerkin method, in particular, some sharp jumping phenomena, which are related to the shock wave, have been captured by the numerical method presented. As the conclusion, it can be drawn that it is possible to completely describe the dynamics on the attractor of a nonlinear partial differential equation (PDE) with a finite-dimensional dynamical system, and the study can provide a numerical method for the analysis of the nonlinear continuous dynamic systems and complicated nonlinear phenomena in finite-dimensional dynamic system, whose nonlinear dynamics has been developed completely compared with infinite-dimensional dynamic system.     

Multidomain Legendre-Galerkin Least-Squares Method for Linear Differential Equations with Variable Coefficients

The multidomain Legendre-Galerkin least-squares method is developed for solving linear differential problems with variable coefficients. By introducing a flux, the original differential equation is rewritten into an equivalent first-order system, and the Legendre Galerkin is applied to the discrete form of the corresponding least squares function. The proposed scheme is based on the Legendre-Galerkin method, and the Legendre/Chebyshev-Gauss-Lobatto collocation method is used to deal with the variable coefficients and the right hand side terms. The coercivity and continuity of the method are proved and the optimal error estimate in $H^1$-norm is obtained. Numerical examples are given to validate the efficiency and spectral accuracy of our scheme. Our scheme is also applied to the numerical solutions of the parabolic problems with discontinuous coefficients and the two-dimensional elliptic problems with piecewise constant coefficients, respectively.     

HIGH-ORDER NONLINEAR GALERKIN METHODS

The nonlinear Galerkin methods are numerical schemes well adapted to the long-term integration of nonlinear evolution partial differential equations. In this paper, a class of high-order nonlinear Galerkin methods are provided. Moreover, convergence results with high-order spectral accuracy are derived for the schemes introduced.     

Computation of nonautonomous invariant and inertial manifolds

We derive a numerical scheme to compute invariant manifolds for time-variant discrete dynamical systems, i.e., nonautonomous difference equations. Our universally applicable method is based on a truncated Lyapunov–Perron operator and computes invariant manifolds using a system of nonlinear algebraic equations which can be solved both locally using (nonsmooth) inexact Newton, and globally using continuation algorithms. Compared to other algorithms, our approach is quite flexible, since it captures time-dependent, nonsmooth, noninvertible or implicit equations and enables us to tackle the full hierarchy of strongly stable, stable and center-stable manifolds, as well as their unstable counterparts. Our results are illustrated using a test example and are applied to a population dynamical model and the Hénon map. Finally, we discuss a linearly implicit Euler–Bubnov–Galerkin discretization of a reaction diffusion equation in order to approximate its inertial manifold.     

ON A CONSTRUCTION OF INERTIAL MANIFOLDS FOR A CLASS OF SECOND ORDER IN TIME EVOLUTION EQUATIONS

Inertial manifolds for a class of second order in time dlssipative equations are constructed.The author also proves an asymptotic completeness property for the inertlal manifolds andcharcterizes the inertlal manifolds as the set of trajectories whose growth is at most of orderO(e-ut) for some u＞O. As applications, a nonllnear wave equation and a problem of nonlinearoscillations of a shallow shell are considered.     

An approximate inertial manifolds approach to postprocessing the Galerkin method for the Navier-Stokes equations

In a recent paper we have introduced a postprocessing procedure for the Galerkin method for dissipative evolution partial differential equations with periodic boundary conditions. The postprocessing technique uses approximate inertial manifolds to approximate the high modes (the small scale components) in the exact solutions in terms of the Galerkin approximations, which in this case play the role of the lower modes (large scale components). This procedure can be seen as a defect-correction technique. But contrary to standard procedures, the correction is computed only when the time evolution is completed. Here we extend these results to more realistic boundary conditions. Specifically, we study in detail the two-dimensional Navier-Stokes equations subject to homogeneous (nonslip) Dirichlet boundary conditions. We also discuss other equations, such as reaction-diffusion systems and the Cahn-Hilliard equations.

     

惯性算法的收敛性和稳定性

1．引言对于非线性发展方程，人们感兴趣的是解的渐近行为．当某一物理参数人很小时，非定常解趋向定常解，而当入充分大时，非定常解的渐近行为完全表现在一个吸引子的结构上，这个吸引子可能是具有分数维数的分形结构．在试图逼近这个吸引子的设想当中，惯性流形显示了它的巨大优越性［1－4］．一个系统的惯性流形是一个光滑的有限维流形，它以指数级速度逼近吸引子．在这个光滑的流形上，一个偏微系统可以用它的惯性形式即有限维常微系统来得到．然而在目前状况下，人们知道存在惯性流形的非线性发展方程为数不多．而绝大部分非线性发展…     

STABILITY ANALYSIS OF NONLINEAR GALERKIN ＆ GALERKIN METHODS FOR NONLINEAR EVOLUTION EQUATIONS

Abstract. Ogr object in this artlcle is to describe tbe Galerkln scheme and nonlin-eax Galerkin scheme for the approximation of nonlinear evolution equations, and tostudy the stability of these schemes. Spatial discretizatlon can be pedormed by eitherGalerkln spectral method or nonlinear Galerldn spectral method; time discretizatlort isdone hy Euler sin.heine wklch is explicit or implicit in the nonlinear terms. According tothe stability analysis of the above schemes, the stability of nonllneex Galerkln methodis better than that of Galexkln method.     

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     

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.     

Generalized Jacobi spectral Galerkin method for fractional pantograph differential equation

This work is concerned with the extension of the Jacobi spectral Galerkin method to a class of nonlinear fractional pantograph differential equations. First, the fractional differential equation is converted to a nonlinear Volterra integral equation with weakly singular kernel. Second, we analyze the existence and uniqueness of solutions for the obtained integral equation. Then, the Galerkin method is used for solving the equivalent integral equation. The error estimates for the proposed method are also investigated. Finally, illustrative examples are presented to confirm our theoretical analysis.     

Approximate Inertial Manifolds of Strongly Damped Nonlinear Wave Equations

In this paper we consider a class of strongly damped nonlinear wave equations. By the transformation of unknown functions and decomposition of operators, we construct a family of approximate inertial manifolds, and obtain the estimate of orders of approximation of such manifolds to solution orbits.     

Inertial manifolds and normal hyperbolicity

Our aim in this article is to derive an existence theorem of inertial manifolds for fairly general equations with a self-adjoint or nonself-adjoint linear operator in a Banach space setting. A sharp form of the spectral gap condition is given. Many other properties are proven including an interesting characterization of the inertial manifold and the normal hyperbolicity of the inertial manifold.     

The approximate manifolds for the generalized (2+1)-dimensional long–short wave equations

In this paper, the long time behavior of the dissipative generalized (2+1)-dimensional long–short wave equations was studied in dynamics. By applying projecting operator and the eigenvalue methods, the approximate inertial manifolds were constructed. And it is proved that arbitrary trajectory of the dissipative generalized (2+1)-dimensional long–short wave equations goes into a small neighborhood of the approximate inertial manifolds after long time.     

APPROXIMATION TO THE GLOBAL ATTRACTOR FOR A NONLINEAR SCHRODINGER EQUATION

Abstract. In the present paper, we deal with the long-time behavior of dissipative partial differenttial equations, and we construct the approximate inertial mardfolds for the nonlbaear Stringer equation with a zero order dlssipation. The order of approximation of these manlfolde to the global attractor is derived.