数学物理学报(英文版)

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2009年, 第29卷, 第6期　刊出日期：2009-11-20 上一期下一期

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WEAKLY COMPRESSIBLE TWO-PRESSURE TWO-PHASE FLOW

Hyeonseong Jin, James Glimm

数学物理学报(英文版). 2009 (6): 1497-1540. DOI: 10.1016/S0252-9602(10)60001-X

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We analyze the limiting behavior of a compressible two-pressure two-phase flow model as the Mach number tends to zero. Formal asymptotic expansions are derived for the solutions of compressible two-phase equations. Expansion coefficients through second order are evaluated in closed form. Underdetermination of incompressible pressures is resolved by information supplied from the weakly compressible theory. The incompressible pressures are uniquely specified by certain details of the compressible fluids from which they are derived as a limit. This aspect of two phase flow in the incompressible limit appears to be new, and results basically from closures which satisfy single phase boundary conditions at the edges of the mixing zone.

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A REACTIVE DYNAMIC CONTINUUM USER EQUILIBRIUM MODEL FOR BI-DIRECTIONAL PEDESTRIAN FLOWS

Yanqun Jiang, Tao Xiong, S.C.Wong, Chi-Wang Shu, Mengping Zhang, Peng Zhang, William H.K. Lam

数学物理学报(英文版). 2009 (6): 1541-1555. DOI: 10.1016/S0252-9602(10)60002-1

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In this paper, a reactive dynamic user equilibrium model is extended to simulate two groups of pedestrians traveling on crossing paths in a continuous walking facility. Each group makes path choices to minimize the travel cost to its destination in a reactive manner based on
instantaneous information. The model consists of a conservation law equation coupled with an Eikonal-type equation for each group. The velocity-density relationship of pedestrian movement is obtained via an experimental method. The model is solved using a finite volume method for the conservation law equation and a fast-marching method for the Eikonal-type equation on unstructured grids. The numerical results verify the rationality of the model and the validity of the numerical method. Based on this continuum model, a number of results, e.g., the formation of strips or moving clusters composed of pedestrians walking to the same destination, are also observed.

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ON GREEN'S FUNCTION FOR HYPERBOLIC-PARABOLIC SYSTEMS

Tai-Ping Liu, Yanni Zeng

数学物理学报(英文版). 2009 (6): 1556-1572. DOI: 10.1016/S0252-9602(10)60003-3

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We study the Green's function for a general hyperbolic-parabolic system, including the Navier-Stokes equations for compressible fluids and the equations for magnetohydrodynamics. More generally, we consider general systems under the basic Kawashima-Shizuta type of
conditions. The first result is to make precise the secondary waves with subscale structure, revealing the nature of coupling of waves
pertaining to different characteristic families. The second result is on the continuous differentiability of the Green's function with
respect to a small parameter when the coefficients of the system are smooth functions of that parameter. The results significantly improve previous results obtained by the authors.

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STRONG COMPACTNESS OF APPROXIMATE SOLUTIONS TO DEGENERATE ELLIPTIC-HYPERBOLIC EQUATIONS WITH DISCONTINUOUS FLUX FUNCTION

Helge Holden, Kenneth H. Karlsen, Darko Mitrovic, Evgueni Yu. Panov

数学物理学报(英文版). 2009 (6): 1573-1612. DOI: 10.1016/S0252-9602(10)60004-5

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Under a non-degeneracy condition on the nonlinearities we show that sequences of approximate entropy solutions of mixed elliptic-hyperbolic equations are strongly precompact in the general case of a Caratheodory flux vector. The proofs are based on deriving localization principles for H-measures associated to sequences of measure-valued functions. This main result implies existence of solutions to degenerate parabolic convection-diffusion equations with discontinuous flux. Moreover, it provides a framework in which one can prove convergence of various types of approximate solutions, such as those generated by the vanishing viscosity method and numerical schemes.

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ASYMPTOTIC RAREFACTION WAVES FOR BALANCE LAWS WITH STIFF SOURCES

W. Lambert, D. Marchesin

数学物理学报(英文版). 2009 (6): 1613-1628. DOI: 10.1016/S0252-9602(10)60005-7

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We study the long time formation of rarefaction waves appearing in balance laws by means of singular perturbation methods. The balance laws are non standard because they contain a variable u that appears only in the flux terms. We present a concrete example occurring in flow of steam, nitrogen and water in porous media and an abstract example for a class of systems of three equations. In the concrete example the zero-order equations resulting from the expansion yield a type of conservation law system called compositional model in Petroleum Engineering. In this work we show how compositional models originate from physically more fundamental systems of balance laws. Under appropriate conditions, we prove that certain solutions of the system of balance laws decay with time to rarefaction wave solutions in the compositional model originating from the system of balance laws.

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A GHOST FLUID BASED FRONT TRACKING METHOD FOR MULTIMEDIUM COMPRESSIBLE FLOWS

王东红, 赵宁, 胡偶, 刘剑明

数学物理学报(英文版). 2009 (6): 1629-1646. DOI: 10.1016/S0252-9602(10)60006-9

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Recent years the modify ghost fluid method (MGFM) and the real ghost fluid method (RGFM) based on Riemann problem have been developed for multimedium compressible flows. According to authors, these methods have only been used with the level set technique to track the interface. In this paper, we combine the MGFM and the RGFM respectively with front tracking method, for which the fluid interfaces are explicitly tracked by connected points. The method is tested with some one-dimensional problems, and its applicability is also studied.
Furthermore, in order to capture the interface more accurately, especially for strong shock impacting on interface, a shock monitor is proposed to determine the initial states of the Riemann problem. The present method is applied to various one-dimensional problems involving strong shock-interface interaction. An extension of the present method to two dimension is also introduced and preliminary results are given.

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THE CARBUNCLE PHENOMENON IS INCURABLE

Volker Elling

数学物理学报(英文版). 2009 (6): 1647-1656. DOI: 10.1016/S0252-9602(10)60007-0

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Numerical approximations of multi-dimensional shock waves sometimes exhibit an instability called the carbuncle phenomenon.
Techniques for suppressing carbuncles are trial-and-error and lack in reliability and generality, partly because theoretical knowledge about carbuncles is equally unsatisfactory. It is not known which numerical schemes are affected in which circumstances, what causes carbuncles to appear and whether carbuncles are purely numerical artifacts or rather features of a continuum equation or model.

This article presents evidence towards the latter: we propose that carbuncles are a special class of entropy solutions which can be physically correct in some circumstances. Using "filaments'', we trigger a single carbuncle in a new and more reliable way, and compute the structure in detail in similarity coordinates. We argue that carbuncles can, in some circumstances, be valid vanishing viscosity limits.
Trying to suppress them is making a physical assumption that may be false.

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SPATIALLY-LOCALIZED SCAFFOLD PROTEINS MAY FACILITATE TO TRANSMIT LONG-RANGE SIGNALS

Xinfeng Liu, Qing Nie

数学物理学报(英文版). 2009 (6): 1657-1669. DOI: 10.1016/S0252-9602(10)60008-2

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Scaffold proteins play an important role in the promotion of signal transmission and specificity during cell signaling. In cells, signaling proteins that make up a pathway are often physically orgnaized into complexes by scaffold proteins[1]. Previous work [2] has shown that
spatial localization of scaffold can enhance signaling locally while simultaneously suppressing signaling at a distance, and the membrane confinement of scaffold proteins may result in a precipitous spatial gradient of the active product protein, high close to the membrane and low within the cell. However, cell-fate decisions critically depend on the temporal pattern of product protein close to the nucleus. In this paper, when phosphorylation signals cannot be transfered by diffusion only, two mechanisms have been proposed for long-range signaling within cells: multiple locations of scaffold proteins and dynamical movement of scaffold proteins. Thus, here we have unveiled how the spatial propagation of the phosphorylated product protein within a cell depends on the spatially and temporal localized scaffold proteins. A class of novel and fast numerical methods for solving stiff reaction diffusion equations with complex domains is briefly introduced.

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A DIAGONAL SPLIT-CELL MODEL FOR THE OVERLAPPING YEE FDTD METHOD

Jinjie Liu, Moysey Brio, Jerome V. Moloney

数学物理学报(英文版). 2009 (6): 1670-1676. DOI: 10.1016/S0252-9602(10)60009-4

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In this paper, we present a nonorthogonal overlapping Yee method for solving Maxwell's equations using the diagonal split-cell model. When material interface is presented, the diagonal split-cell model does not require permittivity averaging so that better accuracy can be achieved.
Our numerical results on optical force computation show that the standard FDTD method converges linearly, while the proposed method achieves quadratic convergence and better accuracy.

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NUMERICAL SIMULATION OF THE XZ TAURI SUPERSONIC ASTROPHYSICAL JET

Carl L. Gardner, Steven J. Dwyer

数学物理学报(英文版). 2009 (6): 1677-1683. DOI: 10.1016/S0252-9602(10)60010-0

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Computational gas dynamical simulations using the WENO-LF method are applied to modeling the high Mach number astrophysical jet XZ Tauri, including the effects of radiative cooling. Mach 55 simulations of the pulsed proto-jet are presented and analyzed in terms of interacting nonlinear waves: terminal Mach disks, bow shocks, and Meshkov-Richtmyer instabilities of the leading jet contact boundary.

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KINETIC FUNCTIONS IN MAGNETOHYDRODYNAMICS WITH RESISTIVITY AND HALL EFFECT

Philippe G. LeFloch, Siddhartha Mishra

数学物理学报(英文版). 2009 (6): 1684-1702. DOI: 10.1016/S0252-9602(10)60011-2

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We consider a nonlinear hyperbolic system of two conservation laws which arises in ideal magnetohydrodynamics and includes second-order
terms accounting for magnetic resistivity and Hall effect. We show that the initial value problem for this model may lead to solutions exhibiting
complex wave structures, including undercompressive nonclassical shock waves. We investigate numerically the subtle competition that takes place between the hyperbolic, diffusive, and dispersive parts of the system. Following Abeyratne, Knowles, LeFloch, and Truskinovsky, who studied similar questions arising in fluid and solid flows, we determine the associated kinetic function which characterizes the dynamics of undercompressive shocks driven by resistivity and Hall effect. To this end, we design a new class of ``schemes with controled dissipation'',
following recent work by LeFloch and Mohammadian. It is now recognized that the equivalent equation associated with a scheme provides a guideline to design schemes that capture physically relevant, nonclassical shocks. We propose a new class of schemes based on high-order entropy conservative, finite differences for the hyperbolic flux, and high-order central differences for the resistivity and Hall terms. These schemes are tested for several regimes of (co-planar or not) initial data and parameter values, and allow us to analyze the properties of nonclassical shocks and establish the existence of monotone kinetic functions in magnetohydrodynamics.

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REVIEW OF TURBULENT MIXING MODELS

Baolian Cheng

数学物理学报(英文版). 2009 (6): 1703-1720. DOI: 10.1016/S0252-9602(10)60012-4

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Fluid mixing is an important phenomenon in many physical applications from supernova explosions to genetic structure formations. In this paper, we overview some theoretical and empirical dynamic mix models, which have been developed over the recent decades, in particular, the ensemble-average micro physical mix model, the multifluid interpenetration mix model, the phenomenological and hybrid turbulent mix models, the buoyancy drag mix model, the single fluid turbulence mix model, and the large eddy simulation mix model. The similarities, distinctions, and connections between these models and their applications are discussed.

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TIME ASYMPTOTIC BEHAVIOR OF THE BIPOLAR NAVIER-STOKES-POISSON SYSTEM

Hai-Liang Li, Tong Yang, Chen Zou

数学物理学报(英文版). 2009 (6): 1721-1736. DOI: 10.1016/S0252-9602(10)60013-6

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The bipolar Navier-Stokes-Poisson system (BNSP) has been used to simulate the transport of charged particles (ions and electrons for instance) under the influence of electrostatic force governed by the self-consistent Poisson equation. The optimal L²time convergence rate for the global classical solution is obtained for a small initial perturbation of the constant equilibrium state. It is shown that due to the electric field, the difference of the charge densities tend to the equilibrium states at the optimal rate (1+t )-^3/4in L²-norm, while the individual momentum of the charged particles converges at the optimal rate (1+t )^-1/4 which is slower than the rate (1+t )^-3/4 for the compressible Navier-Stokes equations (NS). In addition, a new phenomenon on the charge transport is observed regarding the interplay between the two carriers that
almost counteracts the influence of the electric field so that the total density and momentum of the two carriers converges at a faster rate (1+t )^-3/4+εfor any small constant ε > 0. The above estimates reveal the essential difference between the unipolar and the bipolar Navier-Stokes-Poisson systems.

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

Zhiliang Xu, Guang Lin

数学物理学报(英文版). 2009 (6): 1737-1748. DOI: 10.1016/S0252-9602(10)60014-8

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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 non-oscillation 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.

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LINEAR WAVES THAT EXPRESS THE SIMPLEST POSSIBLE PERIODIC STRUCTURE OF THE COMPRESSIBLE EULER EQUATIONS

Blake Temple, Robin Young

数学物理学报(英文版). 2009 (6): 1749-1766. DOI: 10.1016/S0252-9602(10)60015-X

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In this paper we show how the simplest wave structure that balances compression and rarefaction in the nonlinear compressible Euler
equations can be represented in a solution of the linearized compressible Euler equations. Such waves are exact solutions of the
equations obtained by linearizing the compressible Euler equations about the periodic extension of two constant states separated by
entropy jumps. Conditions on the states and the periods are derived which allow for the existence of solutions in the Fourier 1-mode.
In [3, 4, 5] it is shown that these are the simplest linearized waves such that, for almost every period, they are isolated in the kernel of the linearized operator that imposes periodicity, and such that they perturb to nearby nonlinear solutions of the compressible Euler equations that balance compression and rarefaction along characteristics in the formal sense described in [3]. Their fundamental nature thus makes them of interest in their own right.

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FINITE ELEMENT APPROXIMATION OF AN INTEGRO-DIFFERENTIAL OPERATOR

Ding Xia-qi, Luo Pei-zhu

数学物理学报(英文版). 2009 (6): 1767-1776. DOI: 10.1016/S0252-9602(10)60016-1

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This paper deals with the finite element approximation of an integro-differential equation related with Riemann zeta-function.

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