A New Sixth-Order WENO Scheme for Solving Hyperbolic Conservation Laws

In this paper,we develop a new sixth-order WENO scheme by adopting a convex combina-tion of a sixth-order global reconstruction and four low-order local reconstructions.Unlike the classical WENO schemes,the associated linear weights of the new scheme can be any positive numbers with the only requirement that their sum equals one.Further,a very simple smoothness indicator for the global stencil is proposed.The new scheme can achieve sixth-order accuracy in smooth regions.Numerical tests in some one-and two-dimensional bench-mark problems show that the new scheme has a little bit higher resolution compared with the recently developed sixth-order WENO-Z6 scheme,and it is more efficient than the classical fifth-order WENO-JS5 scheme and the recently developed sixth-order WENO6-S scheme.

On the Possible Cometary Nature of the Uchur Cosmic Body(Fall 3.08.1993)

An expeditionary study of the area of the alleged impact event that occurred on 3.08.1993 in the area of the Lower Konkuli River(southeast of the Aldan Highlands,Lurikan Range,Russia)was carried out.According to the materials of remote sensing,the places of collision with the earth of a cosmic body are determined.In the area of the impact of the shock wave on the Earth’s surface,peat samples were selected,the micro probe analysis of which showed the pres­ence of a cosmogenic substance in concentrations 6-8 times higher than the background.Silicate and magnetite micro spheres,native iron,moissanite,and carbon micro tubes coated with a film consisting of pure nickel were found.Of particular interest were the findings of specific Ni film micro structures that allow us to make an assumption about the cometary nature of the Uchur cosmic body.Most researchers associate the observed flights of fireballs with the sub­sequent fall of meteorites.Researchers are trying to find the massive body of the fallen space body.However,often,even after many years of searching,a massive cosmic body cannot be found.This happened when studying the site of the fall of the Tunguska cosmic body.In this case,it remains to be assumed that the cosmic body contained micro­scopic dust particles.The structure and composition of such particles can only be studied using microscopic research methods.When studying the Uchur cosmic body,the authors concluded that it could be of a cometary nature due to the findings of specific particles-thin films of pure nickel on the surface of plant remains of terrestrial origin.This hy­pothesis arose from the recent discovery of atomic nickel vapors in comets.

A Parallel Algorithm for Adaptive Local Refinement of Tetrahedral Meshes Using Bisection

Local mesh refinement is one of the key steps in the implementations of adaptive finite element methods.This paper presents a parallel algorithm for distributed memory parallel computers for adaptive local refinement of tetrahedral meshes using bisection.This algorithm is used in PHG,Parallel Hierarchical Grid (http://lsec.cc.ac.cn/phg/),a toolbox under active development for parallel adaptive finite element solutions of partial differential equations.The algorithm proposed is characterized by allowing simultaneous refinement of submeshes to arbitrary levels before synchronization between submeshes and without the need of a central coordinator process for managing new vertices.Using the concept of canonical refinement, a simple proof of the independence of the resulting mesh on the mesh partitioning is given,which is useful in better understanding the behaviour of the bisectioning refinement procedure.

A NEW NONCONFORMING MIXED FINITE ELEMENT SCHEME FOR THE STATIONARY NAVIER-STOKES EQUATIONS

In this article, a new stable nonconforming mixed finite element scheme is proposed for the stationary Navier-Stokes equations, in which a new low order Crouzeix- Raviart type nonconforming rectangular element is taken for approximating space for the velocity and the piecewise constant element for the pressure. The optimal order error estimates for the approximation of both the velocity and the pressure in L2-norm are established, as well as one in broken H1-norm for the velocity. Numerical experiments are given which are consistent with our theoretical analysis.

Construction and Analysis of Structured Preconditioners for Block Two-by-Two Matrices

For the large sparse block two-by-two real nonsingular matrices, we establish a general framework of structured preconditioners through matrix transformation and matrix approximations. For the specific versions such as modified block Jacobi-type, modified block Gauss-Seidel-type, and modified block unsymmetric (symmetric) Gauss-Seidel-type preconditioners, we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices. Also, we show that when these structured preconditioners are employed to precondition the Krylov subspace methods such as GMRES and restarted GMRES, fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two-by-two coefficient matrices. In particular, these structured preconditioners can lead to high-quality preconditioning matrices for some typical matrices from the real-world applications.

ON THE MONOTONE CONVERGENCE OF THE PROJECTED ITERATION METHODS FOR LINEAR COMPLEMENTARITY PROBLEMS

Under suitable conditions,the monotone convergence about the projected iteration method for solving linear complementarity problem is proved and the influence of the involved parameter matrix on the convergence rate of this method is investigated.

SOME WEIGHT-TYPE HIGH-RESOLUTION DIFFERENCE SCHEMES AND THEIR APPLICATIONS

By the aid of an idea of the weighted ENO schemes, some weight-type high-resolution difference schemes with different orders of accuracy are presented in this paper by using suitable weights instead of the minmod functions appearing in various TVD schemes. Numerical comparisons between the weighted schemes and the non-weighted schemes have been done for scalar equation, one-dimensional Euler equations, two-dimensional Navier-Stokes equations and parabolized Navier-Stokes equations.

An Adaptive Uniaxial Perfectly Matched Layer Method for Time-Harmonic Scattering Problems

The uniaxial perfectly matched layer (PML) method uses rectangular domain to define the PML problem and thus provides greater flexibility and efficiency in deal- ing with problems involving anisotropic scatterers.In this paper an adaptive uniaxial PML technique for solving the time harmonic Helmholtz scattering problem is devel- oped.The PML parameters such as the thickness of the layer and the fictitious medium property are determined through sharp a posteriori error estimates.The adaptive finite element method based on a posteriori error estimate is proposed to solve the PML equa- tion which produces automatically a coarse mesh size away from the fixed domain and thus makes the total computational costs insensitive to the thickness of the PML absorb- ing layer.Numerical experiments are included to illustrate the competitive behavior of the proposed adaptive method.In particular,it is demonstrated that the PML layer can be chosen as close to one wave-length from the scatterer and still yields good accuracy and efficiency in approximating the far fields.

Standard and Economical Cascadic Multigrid Methods for the Mortar Finite Element Methods

In this paper,standard and economical cascadic multigrid methods are considered for solving the algebraic systems resulting from the mortar finite element methods. Both cascadic multigrid methods do not need full elliptic regularity,so they can be used to tackle more general elliptic problems.Numerical experiments are reported to support our theory.

Solving the AKNS Hierarchy by Its Bilinear Form: Generalized Double Wronskian Solutions

通过 Wronskian 技术，一个简单、直接的证明被介绍在双线性的形式的 AKNS 层次概括了两倍 Wronskian 答案。而且由使用一个统一方法，在为它的两倍 Wronskian 形式的 soliton 答案，合理答案， Matveev 答案和 complexitons 被构造。

Finite-Difference Solution of the Helmholtz Equation Based on Two Domain Decomposition Algorithms

In this paper, wave simulation with the finite difference method for the Helmholtz equation based on the domain decomposition method is investigated. The method solves the problem by iteratively solving subproblems defined on smaller subdomains. Two domain decomposition algorithms both for nonoverlapping and overlapping methods are described. More numerical computations including the benchmark Marmousi model show the effectiveness of the proposed algorithms. This method can be expected to be used in the full-waveform inversion in the future.

ACCURACY ENHANCEMENT FOR THE SIGNORINI PROBLEM WITH FINITE ELEMENT METHOD

In this paper,we study the accuracy enhancement for the frictionless Signorini problem on a polygonal domain with linear finite elements.Numerical test is given to verify our result.

Orbital-Free Density Functional Theory for Molecular Structure Calculations

We give here an overview of the orbital-free density functional theory that is used for modeling atoms and molecules.We review typical approximations to the kinetic energy,exchange-correlation corrections to the kinetic and Hartree energies, and constructions of the pseudopotentials.We discuss numerical discretizations for the orbital-free methods and include several numerical results for illustrations.

Conformal Invariant Asymptotic Expansion Approach for Solving （3＋1）-Dimensional JM Equation

(3+1 ) 维的 Jimbo-Miwa (JM ) 方程被使用 Ruan 介绍的保角的不变的 asymptotic 扩大途径近似解决。由解决新(3+1 ) 维的 integrable 当模特儿，它是保角的不变并且拥有 Painleve 性质，近似答案为 JM 方程被获得，包含不仅不仅 one-solitonsolutions 和 multi-soliton 答案。一些近似答案碰巧准确，一些近似答案能由适当地选择在参数之间的关系变得准确。

Full-waveform Velocity Inversion Based on the Acoustic Wave Equation

Full-waveform velocity inversion based on the acoustic wave equation in the time domain is investigated in this paper. The inversion is the iterative minimization of the misfit between observed data and synthetic data obtained by a numerical solution of the wave equation. Two inversion algorithms in combination with the CG method and the BFGS method are described respectively. Numerical computations for two models including the benchmark Marmousi model with complex structure are implemented. The inversion results show that the BFGS-based algorithm behaves better in inversion than the CG-based algorithm does. Moreover, the good inversion result for Marmousi model with the BFGS-based algorithm suggests the quasi-Newton methods can provide an important tool for large-scale velocity inversion. More computations demonstrate the correctness and effectives of our inversion algorithms and code.

Interaction Between Line Soliton and Algebraic Soliton for Asymmetric Nizhnik-Novikov-Veselov Equation

从描述在线 soliton 和代数学的 soliton 之间的相互作用的可变分离途径，代数学的 soliton 解决方案和解决方案开始被选择适当种子答案为获得(2+1 ) 维的 ANNV 方程。相互作用的行为详细被讨论经分解并且图形地。在线 soliton 和代数学的 soliton 之间有二种单个相互作用，这被显示出：1 ) 代数学的 soliton 和线 soliton 宣传并且无穷地坚持的反响的相互作用；2 ) 代数学的 soliton 无穷地分开影响线 soliton 的运动的极其排斥的相互作用。

NEW SECOND ORDER NONCONFORMING TRIANGULAR ELEMENT FOR PLANAR ELASTICITY PROBLEMS

In the use of finite element methods to the planar elasticity problems,one diffculty is to overcome locking when elasticity constant λ→∞.In the case of traction boundary condition,another diffculty is to make the discrete Korn＇s second inequality valid.In this paper,a triangular element is presented.We prove that this element is locking-free,the discrete Korn＇s second inequality holds and the convergence order is two.

UNSTEADY MODEL OF DROP MARANGONI MIGRATION IN MICROGRAVITY

The experiments of drop Marangoni migration have been performed by the drop shift facility of short period of 4.5 s, and the drop accelerates gradually to an asymptotic velocity during the free fall. The unsteady and axisymmetric model is developed to study the drop migration for the case of moderate Reynolds numberRe=O(1), and the results are compared with the experimental ones in the present paper. Both numerical and experimental results show that the migration velocity for moderate Reynolds number is several times smaller than that given by the linear YGB theory.

Generalized Normal Derivatives and Their Applications in DDMs with Nonmatching Grids and DG Methods

A class of normal-like derivatives for functions with low regularity defined on Lipschitz domains are introduced and studied.It is shown that the new normal-like derivatives,which are called the generalized normal derivatives,preserve the major prop- erties of the existing standard normal derivatives.The generalized normal derivatives are then applied to analyze the convergence of domain decomposition methods (DDMs) with nonmatching grids and discontinuous Galerkin (DG) methods for second-order el- liptic problems.The approximate solutions generated by these methods still possess the optimal energy-norm error estimates,even if the exact solutions to the underlying elliptic problems admit very low regularities.

EXPLICIT SQUARE CONSERVING SCHEMES OF LANDAU-LIFSHITZ EQUATIONWITH GILBERT COMPONENT

A kind of explicit square-conserving scheme is proposed for the Landau-Lifshitz equation with Gilbert component. The basic idea was to semidiscrete the Landau-Lifshitz equation into the ordinary differential equations. Then the Lie group method and the Runge-Kutta (RK) method were applied to the ordinary differential equations. The square conserving property and the accuracy of the two methods were compared. Numerical experiment results show the Lie group method has the good accuracy and the square conserving property than the RK method.