非线性欧拉方程组的完美匹配层吸收边界条件

对于非线性Euler方程,提出一类基于完美匹配层(PML)技术的吸收边界条件。首先对线性化的Euler方程设计出PML公式,然后将线性化Euler方程中的通量函数替换成相对应的非线性通量函数,得到非线性的PML方程。考虑到PML方程中包含有一个刚性的源项,文中采用一种隐显Runge-Kutta方法来求解空间半离散后得到的ODE系统。数值实验表明设计的非线性PML吸收边界条件优于传统的特征边界条件。     

On equivariant Euler characteristics

The Euler characteristic of an orbifold M/G as used in string theory is identified with the Euler characteristic of equivariant K -theory K_G(M).     

Preconditioned characteristic boundary conditions for solution of the preconditioned Euler equations at low Mach number flows

Preconditioned characteristic boundary conditions (BCs) are implemented at artificial boundaries for the solution of the two- and three-dimensional preconditioned Euler equations at low Mach number flows. The preconditioned compatibility equations and the corresponding characteristic variables (or the Riemann invariants) based on the characteristic forms of preconditioned Euler equations are mathematically derived for three preconditioners proposed by Eriksson, Choi and Merkle, and Turkel. A cell-centered finite volume Roe’s method is used for the discretization of the preconditioned system of equations on unstructured meshes. The accuracy and performance of the preconditioned characteristic BCs applied at artificial boundaries are evaluated in comparison with the non-preconditioned characteristic BCs and the simplified BCs in computing steady low Mach number flows. The two-dimensional flow over the NACA0012 airfoil and three-dimensional flow over the hemispherical headform are computed and the results are obtained for different conditions and compared with the available numerical and experimental data. The sensitivity of the solution to the size of computational domain and the variation of the angle of attack for each type of BCs is also examined. Indications are that the preconditioned characteristic BCs implemented in the preconditioned system of Euler equations greatly enhance the convergence rate of the solution of low Mach number flows compared to the other two types of BCs.     

Euler Characteristic and Topological Phase Transition of NUT-Kerr-Newman Black Hole

From the Gauss-Bonnet-Chern theorem, the Euler characteristic of NUT-Kerr-Newman black hole is calculated to be some discrete numbers from 0 to 2. We find that the Bekenstein-Hawking entropy is the largest entropy in topology by taking into account of the relationship between the entropy and the Euler characteristic. The NUT-Kerr- Newman black hole evolves from the torus-like topological structure to the spherical structure with the changes of mass, angular momentum, electric and NUT charges. In this process, the Euler characteristic and the entropy are changed discontinuously, which give the topological aspect of the first-order phase transition of NUT-Kerr-Newman black hole. The corresponding latent heat of the topological phase transition is also obtained. The estimated latent heat of the black hole evolving from the star just lies in the range of the energy of gamma ray bursts.     

On the Euler characteristics of random surfaces

In a Monte Carlo computer experiment, we simulate the Gibbs distribution of nonconnected two-dimensional surfaces isometrically embedded in three-dimensional Euclidean space with fixed boundary and the action given by the area. The simulation involves surfaces built out of plaquettes in a cubical lattice. The foam structure is analyzed in terms of correlations of the local fluctuations in the Euler characteristic and the area. The scaling behavior of the area and the Euler characteristic is discussed by varying the boundary. We show evidence of a phase transition point which is independent of the choice of the boundary. An existence proof is given of the thermodynamic limit for the models considered.Supported in part by the Deutsche Forschungsgemeinschaft and NSF grant No. Phy 81 09110A 01.     

Feynman integral relations from parametric annihilators

We study shift relations between Feynman integrals via the Mellin transform through parametric annihilation operators. These contain the momentum space integration by parts relations, which are well known in the physics literature. Applying a result of Loeser and Sabbah, we conclude that the number of master integrals is computed by the Euler characteristic of the Lee–Pomeransky polynomial. We illustrate techniques to compute this Euler characteristic in various examples and compare it with numbers of master integrals obtained in previous works.

     

Absorbing boundary conditions for nonlinear Euler and Navier–Stokes equations based on the perfectly matched layer technique

Absorbing boundary conditions for the nonlinear Euler and Navier–Stokes equations in three space dimensions are presented based on the perfectly matched layer (PML) technique. The derivation of equations follows a three-step method recently developed for the PML of linearized Euler equations. To increase the efficiency of the PML, a pseudo mean flow is introduced in the formulation of absorption equations. The proposed PML equations will absorb exponentially the difference between the nonlinear fluctuation and the prescribed pseudo mean flow. With the nonlinearity in flux vectors, the proposed nonlinear absorbing equations are not formally perfectly matched to the governing equations as their linear counter-parts are. However, numerical examples show satisfactory results. Furthermore, the nonlinear PML reduces automatically to the linear PML upon linearization about the pseudo mean flow. The validity and efficiency of proposed equations as absorbing boundary conditions for nonlinear Euler and Navier–Stokes equations are demonstrated by numerical examples.     

Topological Aspects of Entropy and Phase Transition of Kerr Black Holest

In the light of topological current and the relationship between the entropy and the Euler characteristic, the topological aspects of entropy and phase transition of Kerr black holes are studied. From Gauss-Bonnet-Chern theorem, it is shown that the entropy of Kerr black holes is determined by the singularities of the Killing vector field of spacetime. By calculating the Hopf indices and Brouwer degrees of the Killing vector field at the singularities, the entropy S = A/4 for nonextreme Kerr black holes and S = 0 for extreme ones are obtained, respectively. It is also discussed that, with the change of the ratio of mass to angular momentum for unit mass, the Euler characteristic and the entropy of Kerr black holes will change discontinuously when the singularities on Cauchy horizon merge with the singularities on event horizon, which will lead to the first-order phase transition of Kerr black holes.     

Topological Aspects of Entropy and Phase Transition of Kerr Black Holes

In the light of topological current and the relationship between the entropy and the Euler characteristic, the topological aspects of entropy and phase transition of Kerr black holes are studied. From Gauss-Bonnet-Chern theorem,it is shown that the entropy of Kerr black holes is determined by the singularities of the Killing vector field of spacetime.By calculating the Hopf indices and Brouwer degrees of the Killing vector field at the singularities, the entropy S = A/4for nonextreme Kerr black holes and S = 0 for extreme ones are obtained, respectively. It is also discussed that, with the change of the ratio of mass to angular momentum for unit mass, the Euler characteristic and the entropy of Kerr black holes will change discontinuously when the singularities on Cauchy horizon merge with the singularities on event horizon, which will lead to the first-order phase transition of Kerr black holes.     

Euler characteristic in percolation theory

The notion of the Euler characteristic is introduced in percolation theory, which, in fact, was implicitly used from the very beginning of studying percolation problems. An exact formula is given for the case of the ball problem, along with some of its generalizations.     

A stochastic approach to the Euler-Poincaré number of the loop space of a developable orbifold

We define a regularised version of the de Rham operator over the free loop space. We perform a semi-classical approximation of it, such that the index of the limit operator is equal to the “orbit Euler characteristic” of physicists.     

Stability of Exponential Euler Method for Stochastic Systems under Poisson White Noise Excitations

The stability of stochastic systems under Poisson white noise excitations which based on the quantum theory is investigated in this paper. In general, the exact solution of the most of the stochastic systems with jumps is not easy to get. So it is very necessary to investigate the numerical solution of equations. On the one hand, exponential Euler method is applied to study stochastic delay differential equations, we can find the sufficient conditions for keeping mean square stability by investigating numerical method of systems. Through the comparison, we get the step-size of this method which is longer than the Euler-Maruyama method. On the other hand, mean square exponential stability of exponential Euler method for semi-linear stochastic delay differential equations under Poisson white noise excitations is confirmed.     

Interaction of Four Rarefaction Waves in the Bi-Symmetric Class of the Two-Dimensional Euler Equations

The global existence and structures of solutions to multi-dimensional unsteady compressible Euler equations are interesting and important open problems. In this paper, we construct global classical solutions to the interaction of four orthogonal planar rarefaction waves with two axes of symmetry for the Euler equations in two space dimensions, in the case where the initial rarefaction waves are large. The bi-symmetric initial data is a basic type of four-wave two-dimensional Riemann problems. The solutions in this case are continuous, bounded and self-similar, and we characterize how large the rarefaction waves must be. We use the methods of hodograph transformation, characteristic decomposition, and phase space analysis. We resolve binary interactions of simple waves in the process.     

管道后传声数值模拟的不稳定波抑制

在管道后传声的数值模拟中,必须考虑平均流剪切层的散射效应,然而在非均匀剪切流动下时域求解线化欧拉方程会面临Kelvin-Helmholtz不稳定波产生和放大的难题。已有的不稳定波抑制技术通常很难获得令人满意的结果。本文采用一种混合方法,首先引入有限时段的宽频声源波包将声波和不稳定波分离,进而采用声源滤波器技术对不稳定波进行抑制。数值验证算例选择半无限长轴对称环形硬壁直管道,采用计算气动声学方法时域求解2.5维线化欧拉方程,无背景流动的数值解与解析解符合很好,验证了程序的精度与可靠性,非均匀流动算例则表明所采用波包加声源滤波器混合方法对不稳定波抑制效果明显,对声场影响很小,充分显示了该方法的精度与可行性。     

广义特征坐标系计算膨胀波与激波优越性的数值验证

最近提出的以流体速度和声速的线性组合为速度而运动的广义特征坐标系,是-个更为-般的坐标系统,当参数取不同特殊值的时候,就可以得到其他经典的坐标系统.这种坐标系的最大特点是,它能跟随膨胀波(当然也可以跟随激波和接触间断)运动,从而在使用同-计算格式的前提下,使得膨胀波的计算精度更高.对膨胀波和激波进行数值试验,结果显示广义特征坐标系在计算膨胀波和激波时明显优于传统的坐标系.     

Comments on “Natural frequencies of a uniform cantilever with a tip mass slender in the axial direction”

Using a power series expansion for the roots of the characteristic equation, the natural frequencies of a cantilevered Euler–Bernoulli beam with an end rigid mass are computed. Results are obtained for a range of center of mass offsets from the end of the beam. The results of current study corrects significant errors in the previous work by Rama Bhat and Wagner (1976).     

一种强健的守恒型间断跟踪法在一维Euler方程组上的实现

文[8,9]研究设计了一种守恒型的间断跟踪法,其主要特点是以解的守恒性作为跟踪的机制而不是像传统的间断跟踪法利用Rankine-Hugoniot条件,将该跟踪法对一维Euler方程组进行实现.对一维单个守恒律凸流的实现见文[7].由于Euler方程组有3类特征线,信息沿每类特征线传播并相互穿透.如果仍然象单个守恒律那样间断两侧的数值解的计算只用到来自间断同侧的信息,则间断一侧的信息就不能传到间断的另一侧,而会堆积在间断附近.设计了一种几乎是二阶的按特征方式分离信息的方法,将堆积在间断附近的信息分离.而且按此思想可做成高阶方法.该方法中不仅用到了精确Riemann分解,还用到了Roe^[12]的近似Riemann分解及各种插值和重构技巧.数值实验表明这样的处理是有效的.     

Seiberg--Witten Monopoles in Three Dimensions

Dimensional reduction of the Seiberg--Witten equations leads to the equations of motion of a U(1) Chern--Simons theory coupled to a massless spinorial field. A topological quantum field theory is constructed for the moduli space of gauge equivalence classes of solutions of these equations. The Euler characteristic of the moduli space is obtained as the partition function which yields an analogue of Casson's invariant.A mathematically rigorous definition of the invariant isdeveloped for homology spheres using the theory of spectral flow ofself-adjoint Fredholm operators.     

Topological invariants and the dynamics of an axial vector torsion field

A generalized theory of gravitation is discussed which is based on a Riemann-Cartan space-time,U ₄, with an axial vector torsion field. Besides Einstein's equations determining the metric of theU ₄, a system of nonlinear field equations is established coupling an axial vector source current to the axial vector torsion field. The properties of the solutions of these equations are discussed assuming a London-type condition relating the axial current and torsion field. To characterize the solutions use is made of the Euler and Pontrjagin forms and the associated quadratic curvature invariants for theU ₄ space-time. It is found that there exists for a Riemann-Cartan space-time a relation between the zeros of the axial vector torsion field and the singularities of the Pontrjagin invariant, which is analogous to the well-known Hopf relation between the zeros of vector fields and the Euler characteristic.