高分辨KFVS有限体积方法及其CFD应用

1．引言文中研究三维Euler方程组的数值求解·儿1）中p,（。。,。。,。z）,p和E分别表示流体密度,流体速度矢量,压力和总能．方程组（1．1）是不封闭的,除非增加一个额外的方程一状态方程p一pk句,e表示单位质量内能．本文仅限于理想气体,此时状态方程为p一（、－1加e．队2）近H十年来,涌现了许多求解方程组（1．1）的无振荡、高分辨格式,例如TVD格式问,＊＊O格式问等,它们在一定程度上促进了航空航天和造船事业的发展．其中有一类根据双曲方程组（1．l）特征值的符号建立的迎风格式尤为突出,与中心格式相比,迎风格式的耗…     

动力学的流矢量分裂法模拟浅水波方程

Based on the analogy to gas dynamics, the kinetic flux vector splitting (KFVS) method is used to stimulate the shallow water wave equations. The flux vectors of the equations are split on the basis of the local equilibrium Maxwell-Boltzmann distribution. One dimensional examples including a dam breaking wave and flows over a ridge are calculated. The solutions exhibit second-order accuracy with no spurious oscillation.     

Approximation of liquid–vapor phase transition for compressible fluids with tabulated EOS

This Note investigates the approximation of phase change in compressible fluids with complex equation of state (EOS). Assuming a local and instantaneous equilibrium with respect to phasic pressures, temperatures and chemical potentials when both phases are present leads to the classical definition of an equilibrium EOS for the two-phase medium. Unfortunately, there is no explicit expression of the equilibrium EOS in most cases. We propose simple means to approximate the equilibrium EOS when both phases are governed by very general EOS, including tabulated ones. We present a relaxation type numerical algorithm based on this approximation for simulating two-phase flows with phase change.     

Implicit-explicit schemes for flow equations with stiff source terms

In this paper, we design stable and accurate numerical schemes for conservation laws with stiff source terms. A prime example and the main motivation for our study is the reactive Euler equations of gas dynamics. Furthermore, we consider widely studied scalar model equations. We device one-step IMEX (implicit-explicit) schemes for these equations that treats the convection terms explicitly and the source terms implicitly.For the non-linear scalar equation, we use a novel choice of initial data for the resulting Newton solver and obtain correct propagation speeds, even in the difficult case of rarefaction initial data. For the reactive Euler equations, we choose the numerical diffusion suitably in order to obtain correct wave speeds on under-resolved meshes.We prove that our implicit-explicit scheme converges in the scalar case and present a large number of numerical experiments to validate our scheme in both the scalar case as well as the case of reactive Euler equations.Furthermore, we discuss fundamental differences between the reactive Euler equations and the scalar model equation that must be accounted for when designing a scheme.     

Fluid dynamics and thermodynamics as a unified field theory

We study the problem of consistency of equations of continuum dynamics (using the Euler equations and the continuity equation as examples) and thermodynamic equations of state (for the specific free energy, entropy, and volume). We propose a variant of the Hamiltonian formulation of a model that combines the fluid dynamics of a potential flow of a compressible fluid or gas and local equilibrium thermodynamics into a unified field theory. Thermodynamic equations of state appear in this model as second-class constraint equations. As a consistency condition, there arises another second-class constraint requiring that the product of density and temperature should be independent of time. The model provides an in-principle possibility of finding the time dependence of the specific entropy of the arising dynamical system.     

Computational study of shock waves propagating through air-plastic-water interfaces

The following study ismotivated by experimental studies in traumatic brain injury (TBI). Recent research has demonstrated that low intensity non-impact blast wave exposure frequently leads to mild traumatic brain injury (mTBI); however, the mechanisms connecting the blast waves and the mTBI remain unclear. Collaborators at the Seattle VA Hospital are doing experiments to understand how blast waves can produce mTBI. In order to gain insight that is hard to obtain by experimental means, we have developed conservative finite volume methods for interface-shock wave interaction to simulate these experiments. A 1D model of their experimental setup has been implemented using Euler equations for compressible fluids. These equations are coupled with a Tammann equation of state (EOS) that allows us to model compressible gas along with almost incompressible fluids or elastic solids. A hybrid HLLC-exact Eulerian-Lagrangian Riemann solver for Tammann EOS with a jump in the parameters has been developed. The model has shown that if the plastic interface is very thin, it can be neglected. This result might be very helpful to model more complicated setups in higher dimensions.     

The Runge‐Kutta DG finite element method and the KFVS scheme for compressible flow simulations

This article presents a new type of second‐order scheme for solving the system of Euler equations, which combines the Runge‐Kutta discontinuous Galerkin (DG) finite element method and the kinetic flux vector splitting (KFVS) scheme. We first discretize the Euler equations in space with the DG method and then the resulting system from the method‐of‐lines approach will be discretized using a Runge‐Kutta method. Finally, a second‐order KFVS method is used to construct the numerical flux. The proposed scheme preserves the main advantages of the DG finite element method including its flexibility in handling irregular solution domains and in parallelization. The efficiency and effectiveness of the proposed method are illustrated by several numerical examples in one‐ and two‐dimensional spaces. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Gas-dynamic equations for spatially inhomogeneous gas mixtures with internal degrees of freedom. II. General representation for one-temperature reaction rates

In our previous paper (Kolesnichenko and Gorbachev, 2010) [1] the general approach for solving kinetic equations for gas mixtures with internal degrees of freedom and for getting corresponding gas-dynamic equations was developed. In the present article we continue our studies and focus on formulating expressions for reaction rates arising in zero-order (Euler) gas-dynamic equations for a one-temperature case. Derived expressions take into account all non-equilibrium effects, that we understand as deviation of the distribution function from its quasi-equilibrium value. As was shown in Kolesnichenko and Gorbachev (2010, 2011)  and , for zero-order approximation these effects can be subdivided into two groups. The first group contain the effects caused by the perturbation of quasi-equilibrium distribution function by the physical–chemical processes. We call them “scalar” non-equilibrium effects. Only these effects remain in the spatially homogeneous case. The second group consists of the terms proportional to the velocity divergence and therefor is indicated as spatially inhomogeneous. Both the above-mentioned effects are described via additive corrections to the quasi-equilibrium distribution function. Thus they can be treated separately and give rise to the corresponding terms in the expressions for reaction rates. Those non-equilibrium reaction rates are functions of concentrations of all species presented in the mixture and of the whole set of equilibrium rate constants. This leads to the necessity of developing new approaches to obtaining the reaction rates from experimental data. Traditionally defined an equilibrium constant, that is the function only of the thermodynamic state of the system, can be introduced only in a spatially homogeneous case. It can partially simplify the problem of getting information on reaction rates. In general no such value can be introduced. Final expressions show strong correlations between parallel reactions. One such reaction can affect another one until it vanishes. This can be caused not only by the “scalar” non-equilibrium effects, as it was shown in Kolesnichenko and Gorbachev (2011) [2], but also by spatially inhomogeneous. This means that the derived non-equilibrium terms are not small corrections, but can dramatically change chemical kinetics of the reacting mixture.     

Strong convergence of a stochastic Rosenbrock-type scheme for the finite element discretization of semilinear SPDEs driven by multiplicative and additive noise

This paper aims to investigate the numerical approximation of a general second order parabolic stochastic partial differential equation(SPDE) driven by multiplicative and additive noise. Our main interest is on such SPDEs where the nonlinear part is stronger than the linear part, usually called stochastic dominated transport equations. Most standard numerical schemes lose their good stability properties on such equations, including the current linear implicit Euler method. We discretize the SPDE in space by the finite element method and propose a novel scheme called stochastic Rosenbrock-type scheme for temporal discretization. Our scheme is based on the local linearization of the semi-discrete problem obtained after space discretization and is more appropriate for such equations. We provide a strong convergence of the new fully discrete scheme toward the exact solution for multiplicative and additive noise and obtain optimal rates of convergence. Numerical experiments to sustain our theoretical results are provided.     

Highly Accurate Numerical Schemes for Stochastic Optimal Control via FBSDEs

This work is concerned with numerical schemes for stochastic optimal control problems (SOCPs) by means of forward backward stochastic differential equations (FBSDEs). We first convert the stochastic optimal control problem into an equivalent stochastic optimality system of FBSDEs. Then we design an efficient second order FBSDE solver and an quasi-Newton type optimization solver for the resulting system. It is noticed that our approach admits the second order rate of convergence even when the state equation is approximated by the Euler scheme. Several numerical examples are presented to illustrate the effectiveness and the accuracy of the proposed numerical schemes.     

有限级超越整函数的(微-)差分多项式的零点分布

作者研究了有限级超越整函数的差分多项式和微-差分多项式的零点分布,在一定条件下得到了这些多项式的零点收敛指数的精确估计.所得结果可视为Hayman关于Picard例外值的经典结果的(微-)差分模拟.     

均匀格子气Boltzmann方程的稳定性研究

１．引言格子气的基本方程是在几何空间、速度空间和时间上都是离散的Ｂｏｌｔｚｍａｎｎ方程（Ｂ方程）．这是一个有限差分方程．在离散速度气体运动论中［１］,Ｂ方程在速度空间上是离散的,在几何空间和时间上是连续的．这是一个偏微分方程．人们对离散速度气体Ｂ方程的稳定性和渐近特性的研究已经取得了很多结果．Ｍａａｓｓ~［２］通过构造Ｌｙaｐｕｎｏｖ函数族,在分布函数在空间上均匀的条件下,证明了平衡分布的渐近稳定性．信息函数Ｈ是该函数族的一员．Ｂｅｌｌｏｍｏｅｔａｌ~［３］．采用小扰动线性化方法在初值距离平衡解足…     

Shock waves in the nonisentropic gas flow

In this paper we propose an extended entropy condition for general systems of hyperbolic conservation laws with several space variables. This entropy condition generalizes the well-known condition (E) of Volpert for a single conservation law with several space variables and reduces to the entropy condition proposed earlier by the author for systems with one space variable. The Riemann problem for general nonisentropic gas equations has a unique solution for initial data with arbitrarily large jumps. The occurrence of a vacuum region is observed. The projections of shock curves on the pressure-velocity plane are analyzed so as to study the interaction of weak shocks. Our results differ markedly from those of previous works in that we do not assume the equation of state to be polytropic. In fact our assumptions on the equation of state allow the pressure to be a nonconvex function of specific volume.The Riemann problem for this general system of gas equations was also treated by B. Wendroff when the initial data are near constant.     

Exponential decay in time of solutions of the viscous quantum hydrodynamic equations

The long-time asymptotics of solutions of the viscous quantum hydrodynamic model in one space dimension is studied. This model consists of continuity equations for the particle density and the current density, coupled to the Poisson equation for the electrostatic potential. The equations are a dispersive and viscous regularization of the Euler equations. It is shown that the solutions converge exponentially fast to the (unique) thermal equilibrium state as the time tends to infinity. For the proof, we employ the entropy dissipation method, applied for the first time to a third-order differential equation.     

Product integration methods for solving a system of nonlinear Volterra integral equations

In this paper the technique of subtracting out singularities is used to derive explicit and implicit product Euler schemes with order one convergence and a product trapezoidal scheme with order two convergence for a system of Volterra integral equations with a weakly singular kernel. The convergence proofs of the numerical schemes are presented; these are nonstandard since the nonlinear function involved in the integral equation system does not satisfy a global Lipschitz condition.     

结合气体分子动力学方法的RKDG有限元格式求解一维Euler可压方程组

In this paper, two numerical methods are developed for solving one-dimensionl compressible ELder equations by the RKDG finite element method.The schemesare obtained based on an important relation between the Boltzmann equation andthe ELder equations.The schemes have the TVD-like property under the uniform meshes.Several numerical results also present the performance of the schemes.     

Pointwise structure of a radiation hydrodynamic model in one-dimension

In this paper, we study the nonlinear stability and the pointwise structure around a constant equilibrium for a radiation hydrodynamic model in 1-dimension, in which the behavior of the fluid is described by a full Euler equation with certain radiation effect. It is well-known that the classical solutions of the Euler equation in 1-D may blow up in finite time for general initial data. The global existence of the solution in this paper means that the radiation effect stabilizes the system and prevents the formation of singularity when the initial data is small. To study the precise effect of the radiation in this model, we also treat the pointwise estimates of the solution for the original nonlinear problem by combining the Green's function for the linearized radiation hydrodynamic equations with the Duhamel's principle. The result in this paper shows that the pointwise structure of this model is similar to that of full Navier-Stokes equations in 1-D.     

Modeling of a Hamiltonian conservative chaotic system and its mechanism routes from periodic to quasiperiodic,chaos and strong chaos

In this paper, the important role of 3D Euler equation playing in forced-dissipative chaotic systems is reviewed. In mathematics, rigid-body dynamics, the structure of symplectic manifold, and fluid dynamics, building a four-dimensional (4D) Euler equation is essential. A 4D Euler equation is proposed by combining two generalized Euler equations of 3D rigid bodies with two common axes. In chaos-based secure communications, generating a Hamiltonian conservative chaotic system is significant for its advantage over the dissipative chaotic system in terms of ergodicity, distribution of probability, and fractional dimensions. Based on the proposed 4D Euler equation, a 4D Hamiltonian chaotic system is proposed. Through proof, only center and saddle equilibrium lines exist, hence it is not possible to produce asymptotical attractor generated from the proposed conservative system. An analytic form of Casimir power demonstrates that the breaking of Casimir energy conservation is the key factor that the system produces the aperiodic orbits: quasiperiodic orbit and chaos. The system has strong pseudo-randomness with a large positive Lyapunov exponent (more than 10 K), and a large state amplitude and energy. The bandwidth for the power spectral density of the system is 500 times that of both existing dissipative and conservative systems. The mechanism routes from quasiperiodic orbits to chaos is studied using the Hamiltonian energy bifurcation and Poincaré map. A circuit is implemented to verify the existence of the conservative chaos.     

Stability of viscous contact wave for compressible navier-stokes system of general gas with free boundary

In this paper, we study the large time behavior of solutions to the nonisentropic Navier-Stokes equations of general gas, where polytropic gas is included as a special case, with a free boundary. First we construct a viscous contact wave which approximates to the contact discontinuity, which is a basic wave pattern of compressible Euler equation, in finite time as the heat conductivity tends to zero. Then we prove the viscous contact wave is asymptotic stable if the initial perturbations and the strength of the contact wave are small. This generalizes our previous result [6] which is only for polytropic gas.