CONSERVATION OF THREE—POINT COMPACT SCHEMES ON SINGLE AND MULTIBLOCK PATCHED GRIDS FOR HYPERBOLIC PROBLEMS

For nonlinear hyperbloic problems,Conservation of the numerical scheme is important for convergence to the correct weak solutions.In this paper the the conservation of the well-known compact scheme up to fourth order of accuracy on a single and uniform grid is studied,and a conservative interface treatment is derived for compact schemes on patched grids .For a pure initial value problem,the compact scheme is shown to be equivalent to a scheme in the usual conservative form .For the case of a mixed initial boundary value problem,the compact scheme is conservative only if the rounding errors are small enough.For a pactched grid interface,a conservative interface condition useful for mesh fefiement and for parallel computation is derived and its order of local accuracy is analyzed.     

AN EFFICIENT MOVING MESH METHOD FOR A MODEL OF TURBULENT FLOW IN CIRCULAR TUBES

This paper presents an efficient moving problem with an optimal control constrained mesh method to solve a nonlinear singular condition. The physical problem is governed by a new model of turbulent flow in circular tubes proposed by Luo et al. using Prandtl＇s mixing-length theory. Our algorithm is formed by an outer iterative algorithm for handling the optimal control condition and an inner adaptive mesh redistribution algorithm for solving the singular governing equations. We discretize the nonlinear problem by using a upwinding approach, and the resulting nonlinear equations are solved by using the Newton- Raphson method. The mesh is generated and the grid points are moved by using the arc-length equidistribution principle. The numerical results demonstrate that proposed algorithm is effective in capturing the boundary layers associated with the turbulent model.     

ON NON-ISOTROPIC JACOBI PSEUDOSPECTRAL METHOD

In this paper, a non-isotropic Jacobi pseudospectral method is proposed and its appli- cations are considered. Some results on the multi-dimensional Jacobi-Gauss type interpolation and the related Bernstein-Jackson type inequalities are established, which play an important role in pseudospectral method. The pseudospectral method is applied to a twodimensional singular problem and a problem on axisymmetric domain. The convergence of proposed schemes is established. Numerical results demonstrate the efficiency of the proposed method.     

Calculating the vertex unknowns of nine point scheme on quadrilateral meshes for diffusion equation

In the construction of nine point scheme,both vertex unknowns and cell-centered unknowns are introduced,and the vertex unknowns are usually eliminated by using the interpolation of neighboring cell-centered unknowns,which often leads to lose accuracy.Instead of using interpolation,here we propose a different method of calculating the vertex unknowns of nine point scheme,which are solved independently on a new generated mesh.This new mesh is a Vorono¨i mesh based on the vertexes of primary mesh and some additional points on the interface.The advantage of this method is that it is particularly suitable for solving diffusion problems with discontinuous coeffcients on highly distorted meshes,and it leads to a symmetric positive definite matrix.We prove that the method has first-order convergence on distorted meshes.Numerical experiments show that the method obtains nearly second-order accuracy on distorted meshes.     

A Second-Order Method for the Electromagnetic Scattering from a Large Cavity

In this paper,we study the electromagnetic scattering from a two dimen- sional large rectangular open cavity embedded in an infinite ground plane,which is modelled by Helmholtz equations.By introducing nonlocal transparent boundary con- ditions,the problem in the open cavity is reduced to a bounded domain problem.A hypersingular integral operator and a weakly singular integral operator are involved in the TM and TE cases,respectively.A new second-order Toeplitz type approximation and a second-order finite difference scheme are proposed for approximating the hyper- singular integral operator on the aperture and the Helmholtz in the cavity,respectively. The existence and uniqueness of the numerical solution in the TE case are established for arbitrary wavenumbers.A fast algorithm for the second-order approximation is pro- posed for solving the cavity model with layered media.Numerical results show the second-order accuracy and efficiency of the fast algorithm.More important is that the algorithm is easy to implement as a preconditioner for cavity models with more general media.     

TWO-LEVEL MULTISCALE FINITE ELEMENT METHODS FOR THE STEADY NAVIER-STOKES PROBLEM

In this article, on the basis of two-level discretizations and multiscale finite element method, two kinds of finite element algorithms for steady Navier-Stokes problem are presented and discussed. The main technique is first to use a standard finite element discretization on a coarse mesh to approximate low frequencies, then to apply the simple and Newton scheme to linearize discretizations on a fine grid. At this process, multiscale finite element method as a stabilized method deals with the lowest equal-order finite element pairs not satisfying the inf-sup condition. Under the uniqueness condition, error analyses for both algorithms are given. Numerical results are reported to demonstrate the effectiveness of the simple and Newton scheme.     

ROBUST HIGH ORDER CONVERGENCE OF AN OVERLAPPING SCHWARZ METHOD FOR SINGULARLY PERTURBED SEMILINEAR REACTION-DIFFUSION PROBLEMS

In this article we propose an overlapping Schwarz domain decomposition method for solving a singularly perturbed semilinear reaction-diffusion problem. The solution to this problem exhibits boundary layers of width O（√ε ln（1/√ε）） at both ends of the domain due to the presence of singular perturbation parameter ε. The method splits the domain into three overlapping subdomains, and uses the Numerov or Hermite scheme with a uniform mesh on two boundary layer subdomains and a hybrid scheme with a uniform mesh on the interior subdomain. The numerical approximations obtained from this method are proved to be almost fourth order uniformly convergent （in the maximum norm） with respect to the singular perturbation parameter. Furthermore, it is proved that, for small ε, one iteration is sufficient to achieve almost fourth order uniform convergence. Numerical experiments are given to illustrate the theoretical order of convergence established for the method.     

THE GLOBAL ARTIFICIAL BOUNDARY CONDITIONS FOR NUMERICAL SIMULATIONS OF THE 3D FLOW AROUND A SUBMERGED BODY

We consider the numerical approximations of the three-dimensional steady potential flow around a body moving in a liquid of finite constant depth at constant speed and distance below a free surface in a channel. One vertical side is introduced as the up-stream artificial boundary and two vertical sides are introduced as the downstream arti-ficial boundaries. On the artificial boundaries, a sequence of high-order global artificial boundary conditions are given. Then the original problem is reduced to a problem defined on a finite computational domain, which is equivalent to a variational problem. After solving the variational problem by the finite element method, we obtain the numerical approximation of the original problem. The numerical examples show that the artificial boundary conditions given in this paper are very effective.     

NATURAL BOUNDARY ELEMENT METHOD FOR THREE DIMENSIONAL EXTERIOR HARMONIC PROBLEM WITH AN INNER PROLATE SPHEROID BOUNDARY

In this paper, we study natural boundary reduction for Laplace equation with Dirichletor Neumann boundary condition in a three-dimensional unbounded domain, which is theoutside domain of a prolate spheroid. We express the Poisson integral formula and naturalintegral operator in a series form explicitly. Thus the original problem is reduced to aboundary integral equation on a prolate spheroid. The variational formula for the reducedproblem and its well-posedness are discussed. Boundary element approximation for thevariational problem and its error estimates, which have relation to the mesh size andthe terms after the series is truncated, are also presented. Two numerical examples arepresented to demonstrate the effectiveness and error estimates of this method.     

TWO-LEVEL METHOD FOR UNSTEADY NAVIER-STOKES EQUATIONS IN STREAM FUNCTION FORM

Two-level finite element approximation to stream function form of unsteady Navier-Stokes equations is studied. This algorithm involves solving one nonlinear system on a coarse grid and one linear problem on a fine grid. Moreover,the scaling between these two grid sizes is super-linear. Approximation,stability and convergence aspects of a fully discrete scheme are analyzed. At last a numrical example is given whose results show that the algorithm proposed in this paper is effcient.     

CONSTRUCTION OF GEOMETRIC PARTIAL DIFFERENTIAL EQUATIONS FOR LEVEL SETS

Geometric partial differential equations of level-set form are usually constructed by a variational method using either Dirac delta function or co-area formula in the energy functional to be minimized. However, the equations derived by these two approaches are not consistent. In this paper, we present a third approach for constructing the level-set form equations. By representing various differential geometry quantities and differential geometry operators in terms of the implicit surface, we are able to reformulate three classes of parametric geometric partial differential equations （second-order, fourth-order and sixth- order） into the level-set forms. The reformulation of the equations is generic and simple, and the resulting equations are consistent with their parametric form counterparts. We further prove that the equations derived using co-area formula are also consistent with the parametric forms. However, these equations are of much complicated forms than these given by the equations we derived.     

A LIL and limit distributions for trimmed sums of random vectors attracted to operator semi-stable laws

Let θ∈ Rdbe a unit vector and let X,X1,X2,...be a sequence of i.i.d.Rd-valued random vectors attracted to operator semi-stable laws.For each integer n ≥ 1,let X1,n ≤···≤ Xn,n denote the order statistics of X1,X2,...,Xn according to priority of index,namely | X1,n,θ | ≥···≥ | Xn,n,θ |,where ·,· is an inner product on Rd.For all integers r ≥ 0,define by(r)Sn = n-ri=1Xi,n the trimmed sum.In this paper we investigate a law of the iterated logarithm and limit distributions for trimmed sums(r)Sn.Our results give information about the maximal growth rate of sample paths for partial sums of X when r extreme terms are excluded.A stochastically compactness of(r)Sn is obtained.     

STRONG APPROXIMATION FOR MOVING AVERAGE PROCESSES UNDER DEPENDENCE ASSUMPTIONS

Let {Xt,t ≥ 1} be a moving average process defined by Xt = ∑^∞ k=0 αkξt-k, where {αk,k ≥ 0} is a sequence of real numbers and {ξt,-∞ 〈 t 〈 ∞} is a doubly infinite sequence of strictly stationary dependent random variables. Under the conditions of {αk, k ≥ 0} which entail that {Xt, t ≥ 1} is either a long memory process or a linear process, the strong approximation of {Xt, t ≥ 1} to a Gaussian process is studied. Finally, the results are applied to obtain the strong approximation of a long memory process to a fractional Brownian motion and the laws of the iterated logarithm for moving average processes.     

A remark on strong law of large numbers for weighted U-statistics

Let {X,Xn; n ≥ 1} be a sequence of i.i.d.random variables with values in a measurable space(S,S) such that E|h(X1,X2,...,Xm)| ∞,where h is a measurable symmetric function from Sminto R =(-∞,∞).Let {wn,i1,i2,...,im; 1 ≤ i1 i2 ··· im ≤ n,n ≥ m} be a matrix array of real numbers.Motivated by a result of Choi and Sung(1987),in this note we are concerned with establishing a strong law of large numbers for weighted U-statistics with kernel h of degree m.We show that lim n→∞m!(n-m)!n!1≤i1i2···im≤n wn,i1,i2,...,im(h(Xi1,Xi2,...,Xim)-θ)=0 a.s.whenever supn≥mmax1≤i1i2···im≤n|wn,i1,i2,...,im|∞,whereθ=Eh(X1,X2,...,Xm).The proof of this result is based on a new general result on complete convergence,which is a fundamental tool,for array of real-valued random variables under some mild conditions.     

Equation of motion for a spherically-symmetric shell in the relativistic theory of gravity

In the framework of the relativistic theory of gravity, the equation of motion for a spherically-symmetric singular shell is derived and integrated in the first approximation of the Newton potential U = m/r. We use the covariant energy-momentum conservation law for matter in the effective Riemannian space and, independently, the energy-momentum conservation law for the matter + gravity system in Minkowski space. For the problem under consideration, we show the equivalence of our approach to the classical formalism of singular shells.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 2, pp. 344–352, May, 1996.Translated by A. M. Semikhatov.     

A two-dimensional glimm type scheme on cauchy problem of two-dimensional scalar conservation law

In this paper,we construct a new two-dimensional convergent scheme to solve Cauchy problem of following two-dimensional scalar conservation law{?_tu +?_xf(u) + ?_yg(u) = 0,u(x,y,0) = u_0(x,y).In which initial data can be unbounded.Although the existence and uniqueness of the weak entropy solution are obtained,little is known about how to investigate two-dimensional or higher dimensional conservation law by the schemes based on wave interaction of 2D Riemann solutions and their estimation.So we construct such scheme in our paper and get some new results.     

Structure of entropy solutions to the eikonal equation

In this paper, we establish rectifiability of the jump set of an S ¹–valued conservation law in two space–dimensions. This conservation law is a reformulation of the eikonal equation and is motivated by the singular limit of a class of variational problems. The only assumption on the weak solutions is that the entropy productions are (signed) Radon measures, an assumption which is justified by the variational origin. The methods are a combination of Geometric Measure Theory and elementary geometric arguments used to classify blow–ups.?The merit of our approach is that we obtain the structure as if the solutions were in BV, without using the BV–control, which is not available in these variationally motivated problems. Received June 24, 2002 / final version received November 12, 2002?Published online February 7, 2003     

SPECTRAL/HP ELEMENT METHOD WITH HIERARCHICAL RECONSTRUCTION FOR SOLVING NONLINEAR HYPERBOLIC CONSERVATION LAWS

The hierarchical reconstruction （HR） [Liu, Shu, Tadmor and Zhang, SINUM ＇07] has been successfully applied to prevent oscillations in solutions computed by finite volume, Runge-Kutta discontinuous Galerkin, spectral volume schemes for solving hyperbolic conservation laws. In this paper, we demonstrate that HR can also be combined with spectral/hp element method for solving hyperbolic conservation laws. An orthogonal spectral basis written in terms of Jacobi polynomials is applied. High computational efficiency is obtained due to such matrix-free algorithm. The formulation is conservative, and essential nomoscillation is enforced by the HR limiter. We show that HR preserves the order of accuracy of the spectral/hp element method for smooth solution problems and generate essentially non-oscillatory solutions profiles for capturing discontinuous solutions without local characteristic decomposition. In addition, we introduce a postprocessing technique to improve HR for limiting high degree numerical solutions.     

PREDUAL SPACES FOR Q SPACES

To find the predual spaces Pα（R^n） of Qα（R^n） is an important motivation in the study of Q spaces. In this article, wavelet methods are used to solve this problem in a constructive way. First, an wavelet tent atomic characterization of Pα（Rn） is given, then its usual atomic characterization and Poisson extension characterization are given. Finally, the continuity on Pα of Calderon-Zygmund operators is studied, and the result can be also applied to give the Morrey characterization of Pα（Rn）.     

A TWO-SCALE HIGHER-ORDER FINITE ELEMENT DISCRETIZATION FOR SCHROEDINGER EQUATION

In this paper, a two-scale higher-order finite element discretization scheme is proposed and analyzed for a Schroedinger equation on tensor product domains. With the scheme, the solution of the eigenvalue problem on a fine grid can be reduced to an eigenvalue problem on a much coarser grid together with some eigenvalue problems on partially fine grids. It is shown theoretically and numerically that the proposed two-scale higher-order scheme not only significantly reduces the number of degrees of freedom but also produces very accurate approximations.