Finite difference scheme based on proper orthogonal decomposition for the nonstationary Navier-Stokes equations

The proper orthogonal decomposition(POD)and the singular value decomposition(SVD) are used to study the finite difference scheme(FDS)for the nonstationary Navier-Stokes equations. Ensembles of data are compiled from the transient solutions computed from the discrete equation system derived from the FDS for the nonstationary Navier-Stokes equations.The optimal orthogonal bases are reconstructed by the elements of the ensemble with POD and SVD.Combining the above procedures with a Galerkin projection approach yields a new optimizing FDS model with lower dimensions and a high accuracy for the nonstationary Navier-Stokes equations.The errors between POD approximate solutions and FDS solutions are analyzed.It is shown by considering the results obtained for numerical simulations of cavity flows that the error between POD approximate solution and FDS solution is consistent with theoretical results.Moreover,it is also shown that this validates the feasibility and efficiency of POD method.     

平面不可压缩的Navier-Stokes方程的一个新七模截断(英文)

A new seven-modes truncation of Fourier series of Navier-Stokes equations for a two-dimensional incompressible fluid on a torus is obtained.And its stationary solutions,the existence of attractor and the global stability of the equations are firmly proved.At the same time,several issues such as some basic dynamical behaviors and routs to chaos are shown numerically by changing Reynolds number.The system exhibits a stochastic behavior approached through an involved sequence of bifurcations.     

Unconditionally Optimal Error Estimates of the Bilinear-Constant Scheme for Time-Dependent Navier-Stokes Equations

In this paper,the unconditional error estimates are presented for the time-dependent Navier-Stokes equations by the bilinear-constant scheme.The corresponding optimal error estimates for the velocity and the pressure are derived unconditionally,while the previous works require certain time-step restrictions.The analysis is based on an iterated time-discrete system,with which the error function is split into a temporal error and a spatial error.The τ-independent(τ is the time stepsize)error estimate between the numerical solution and the solution of the time-discrete system is proven by a rigorous analysis,which implies that the numerical solution in L^∞-norm is bounded.Thus optimal error estimates can be obtained in a traditional way.Numerical results are provided to confirm the theoretical analysis.     

Navier-Stokes方程的非奇异解分支的谱Galerkin逼近

No error estimate of the spectral Galerkin approximation for the steady-state Navier-Stokes equations was given without assuming that the data of the externalforce field and the boundary conditions are small enough. In this paper, under the condition that the solutions of the Navier-Stokes equations are nonsingular,we proved the existence and convergence of the spectral Galerkin approximation solutions and gave the error estimate. At last, this approximation method wasapplied to simulate the spherical Couette flow.     

Global Attractors and Determining Modes for the 3D Navier-Stokes-Voight Equations

The authors investigate the long-term dynamics of the three-dimensional Navier- Stokes-Voight model of viscoelastic incompressible fluid. Specifically, upper bounds for the number of determining modes are derived for the 3D Navier-Stokes-Voight equations and for the dimension of a global attractor of a semigroup generated by these equations. Viewed from the numerical analysis point of view the authors consider the Navier-Stokes-Voight model as a non-viscous （inviscid） regularization of the three-dimensional Navier-Stokes equations. Furthermore, it is also shown that the weak solutions of the Navier-Stokes- Voight equations converge, in the appropriate norm, to the weak solutions of the inviscid simplified Bardina model, as the viscosity coefficient v →0.     

STABLE FOURTH-ORDER STREAM-FUNCTION METHODS FOR INCOMPRESSIBLE FLOWS WITH BOUNDARIES

Fourth-order stream-function methods are proposed for the time dependent, incom- pressible Navier-Stokes and Boussinesq equations. Wide difference stencils are used instead of compact ones and the boundary terms are handled by extrapolating the stream-function values inside the computational domain to grid points outside, up to fourth-order in the noslip condition. Formal error analysis is done for a simple model problem, showing that this extrapolation introduces numerical boundary layers at fifth-order in the stream-function. The fourth-order convergence in velocity of the proposed method for the full problem is shown numerically.     

Dimension splitting method for the three dimensional rotating Navier-Stokes equations

In this paper, we propose a dimensional splitting method for the three dimensional (3D) rotating Navier-Stokes equations. Assume that the domain is a channel bounded by two surfaces and is decomposed by a series of surfaces ■i into several sub-domains, which are called the layers of the flow. Every interface i between two sub-domains shares the same geometry. After establishing a semi-geodesic coordinate (S-coordinate) system based on ■i , Navier-Stoke equations in this coordinate can be expressed as the sum of two operators, of which one is called the membrane operator defined on the tangent space on ■i , another one is called the bending operator taking value in the normal space on ■i . Then the derivatives of velocity with respect to the normal direction of the surface are approximated by the Euler central difference, and an approximate form of Navier-Stokes equations on the surface ■i is obtained, which is called the two-dimensional three-component (2D-3C) Navier-Stokes equations on a two dimensional manifold. Solving these equations by alternate iteration, an approximate solution to the original 3D Navier-Stokes equations is obtained. In addition, the proof of the existence of solutions to 2D-3C Navier-Stokes equations is provided, and some approximate methods for solving 2D-3C Navier-Stokes equations are presented.     

Mean field limit of a dynamical model of polymer systems

This paper provides a mathematically rigorous foundation for self-consistent mean feld theory of the polymeric physics.We study a new model for dynamics of mono-polymer systems.Every polymer is regarded as a string of points which are moving randomly as Brownian motions and under elastic forces.Every two points on the same string or on two diferent strings also interact under a pairwise potential V.The dynamics of the system is described by a system of N coupled stochastic partial diferential equations(SPDEs).We show that the mean feld limit as N→∞of the system is a self-consistent McKean-Vlasov type equation,under suitable assumptions on the initial and boundary conditions and regularity of V.We also prove that both the SPDE system of the polymers and the mean feld limit equation are well-posed.     

On the Navier-Stokes Equations for Exothermically Reacting Compressible Fluids

Abstract We analyze mathematical models governing planar flow of chemical reaction from unburnt gasesto burnt gases in certain physical regimes in which diffusive effects such as viscosity and heat conduction aresignificant. These models can be then formulated as the Navier-Stokes equations for exothermically reactingcompressible fluids. We first establish the existence and dynamic behavior, including stability, regularity, andlarge-time behavior, of global discontinuous solutions of large oscillation to the Navier-Stokes equations withconstant adiabatic exponent γ and specific heat C_v. Our approach for the existence and regularity is to combinethe difference approximation techniques with the energy methods, total variation estimates, and weak conver-gence argumeots to deal with large jump discontinuities; and for large-time behavior is an a posteriori argumentdirectly from the weak form of the equations. The approach and ideas we develop here can be applied to solvinga more complicated model where γ     

关于非定常不可压Navier-Stokes方程的时间高精度隐式差分方法

The incompressible Navier-Stokes equations,upon spatial discretization,become a system of differential algebraic equations,formally of index2.But due to the special forms of the discrete gradient and disrete divergence,its index can be regarded as 1.Thus,in this paper,a systematic approach following the ODE theory and methods is presented for the construction of high-order time-accurate implicit schemes for the incompressible Navier-Stokes equations,with projection methods for efficiency of numerical solution.The 3rd order 3-step BDF with componentconsistent pressure-correction projection method is a first attempt in this direction;the related iterative solution of the auxiliary velocyty,the boundary conditions and the stability of the algorithm are discussed.Results of numerical tests on the incompressible Navier-Stokes equations with an exact solution are presented,confirming the accureacy,stability and component-consistency of the proposed method.     

Global asymptotical stability for a fishery model with seasonal harvesting

A new fishery model is proposed by using the strategy of seasonal harvesting. Sufficient and necessary conditions are established to ensure the existence of a unique equilibrium or a periodic solution by the approach of Poincar′e maps. It is shown that the equilibrium or the periodic solution is globally asymptotically stable. Numerical examples are provided to demonstrate the model dynamics and some biological implications are given as well.     

Scaling limits of interacting diffusions in domains

We survey recent effort in establishing the hydrodynamic limits and the fluctuation limits for a class of interacting diffusions in domains. These systems are introduced to model the transport of positive and negative charges in solar cells. They are general microscopic models that can be used to describe macroscopic phenomena with coupled boundary conditions, such as the popula- tion dynamics of two segregated species under competition. Proving these two types of limits represents establishing the functional law of large numbers and the functional central limit theorem, respectively, for the empirical measures of the spatial positions of the particles. We show that the hydrodynamic limit is a pair of deterministic measures whose densities solve a coupled nonlinear heat equations, while the fluctuation limit can be described by a Gaussian Markov process that solves a stochastic partial differential equation.     

AN ANISOTROPIC NONCONFORMING FINITE ELEMENT METHOD FOR APPROXIMATING A CLASS OF NONLINEAR SOBOLEV EQUATIONS

An anisotropic nonconforming finite element method is presented for a class of nonlinear Sobolev equations. The optimal error estimates and supercloseness are obtained for both semi-discrete and fully-discrete approximate schemes, which are the same as the traditional finite element methods. In addition, the global superconvergence is derived through the postprocessing technique. Numerical experiments are included to illustrate the feasibility of the proposed method.     

Zero dissipation limit to a Riemann solution consisting of two shock waves for the 1D compressible isentropic Navier-Stokes equations

We investigate the zero dissipation limit problem of the one-dimensional compressible isentropic Navier-Stokes equations with Riemann initial data in the case of the composite wave of two shock waves.It is shown that the unique solution to the Navier-Stokes equations exists for all time,and converges to the Riemann solution to the corresponding Euler equations with the same Riemann initial data uniformly on the set away from the shocks,as the viscosity vanishes.In contrast to previous related works,where either the composite wave is absent or the efects of initial layers are ignored,this gives the frst mathematical justifcation of this limit for the compressible isentropic Navier-Stokes equations in the presence of both composite wave and initial layers.Our method of proof consists of a scaling argument,the construction of the approximate solution and delicate energy estimates.     

A MODIFIED TWO-DIMENSIONAL LOTKA-VOLTERRA MODEL WITH EXCHANGE BETWEEN COOPERATION AND COMPETITION

Under the assumptions that there exists exchange between cooperation and competition in some ecological systems, the paper provides a modei based on the Lotka-Volterra equations to describe the exchange in a two-dimensional system. Then dynamics of the model is studied in detail, the division of distinct regions associated with difFerent dynamics in the parameter space is obtained, and the biological significance of these dynamical properties represent is discussed.     

Non-gaussian Test Models for Prediction and State Estimation with Model Errors

Turbulent dynamical systems involve dynamics with both a large dimensional phase space and a large number of positive Lyapunov exponents. Such systems are ubiquitous in applications in contemporary science and engineering where the statistical ensemble prediction and the real time filtering/state estimation are needed despite the underlying complexity of the system. Statistically exactly solvable test models have a crucial role to provide firm mathematical underpinning or new algorithms for vastly more complex scientific phenomena. Here, a class of statistically exactly solvable non-Gaussian test models is introduced, where a generalized Feynman-Kac formulation reduces the exact behavior of conditional statistical moments to the solution to inhomogeneous Fokker-Planck equations modified by linear lower order coupling and source terms. This procedure is applied to a test model with hidden instabilities and is combined with information theory to address two important issues in the contemporary statistical prediction of turbulent dynamical systems: the coarse-grained ensemble prediction in a perfect model and the improving long range forecasting in imperfect models. The models discussed here should be useful for many other applications and algorithms for the real time prediction and the state estimation.     

Optimal Delaunay Triangulations

The Delaunay triangulation, in both classic and more generalized sense, is studied in this paper for minimizing the linear interpolation error (measure in L^P-norm) for a given function. The classic Delaunay triangulation can then be characterized as an optimal triangulation that minimizes the interpolation error for the isotropic function ‖x‖^2 among all the triangulations with a given set of vertices. For a more general function, a functiondependent Delaunay triangulation is then defined to be an optimal triangulation that minimizes the interpolation error for this function and its construction can be obtained by a simple lifting and projection procedure. The optimal Delaunay triangulation is the one that minimizes the interpolation error among all triangulations with the same number of vertices, i.e. the distribution of vertices are optimized in order to minimize the interpolation error. Such a function-depend entoptimal Delaunay triangulation is proved to exist for any given convex continuous function.On an optimal Delaunay triangulation associated with f, it is proved that △↓f at the interior vertices can be exactly recovered by the function values on its neighboring vertices.Since the optimal Delaunay triangulation is difficult to obtain in practice, the concept of nearly optimal triangulation is introduced and two sufficient conditions are presented for a triangulation to be nearly optimal.     

SIGN MATRIX, REGULAR SPLITTINGS AND MONOTONIC ENCLOSURE OF SOLUTIONS FOR NONLINEAR SYSTEM OF EQUATIONS, PARTI

In this paper we introduce the sign matrix of a nonlinear system of equations x = Gx to characterize its hybrid and asynchronous monotonicity as well as convexity. Based on the configuration of the matrix, we define a new type of regular splittings of the system with which the solvability and construction of solutions for the system are transformed to those of the couple systems of the splitting formIt is shown that this couple systems is a general model for developing monotonic enclosure methods of solutions for various types of nonlinear system of equations.     

Trajectory Attractors for Binary Fluid Mixtures in 3D

Two different models for the evolution of incompressible binary fluid mixtures in a three-dimensional bounded domain are considered.They consist of the 3D incompressible Navier-Stokes equations,subject to time-dependent external forces and coupled with either a convective Allen-Cahn or Cahn-Hilliard equation.Such systems can be viewed as generalizations of the Navier-Stokes equations to two-phase fluids.Using the trajectory approach,the authors prove the existence of the trajectory attractor for both systems.