Adaptive mesh refinement for stochastic reaction–diffusion processes

We present an algorithm for adaptive mesh refinement applied to mesoscopic stochastic simulations of spatially evolving reaction–diffusion processes. The transition rates for the diffusion process are derived on adaptive, locally refined structured meshes. Convergence of the diffusion process is presented and the fluctuations of the stochastic process are verified. Furthermore, a refinement criterion is proposed for the evolution of the adaptive mesh. The method is validated in simulations of reaction–diffusion processes as described by the Fisher–Kolmogorov and Gray–Scott equations.     

A mechanical model of Brownian motion in half-space

We consider the motion of a heavy mass in an ideal gas in a semi-infinite system, with elastic collisions at the boundary. The motion is determined by elastic collisions. We prove in the Brownian motion limit the convergence of the position and velocity process of the heavy particle to a diffusion process in which velocity and position remain coupled.     

Influence of atomic defect generation on the propagation of elastic waves in laser-excited solid layers

The mechanical behavior of solid layers subjected to laser irradiation is investigated by a dynamical model that is based on coupled evolution equations for the elastic displacement of the medium and lattice defect-density fields. The evolution of defect-density is governed by the (i) generation of defects by irradiation, (ii) their diffusion and recombination and (iii) diffusion induced by strain field. The strain field associated with lattice dilatation due to atomic defects is shown to couple with deformation fields of the layer. Frequency equations corresponding to the symmetric and anti-symmetric modes of vibration of the layer are obtained. It is found that coupling between diffusion and strain fields cause dispersion of the general waveform. Explicit expressions are defined for the wave velocity, and the attenuation (amplification) coefficients which characterize these waves.     

Advection of passive particles in the Kolmogorov flow

A statistical analysis of the advection of passive particles in a flow governed by driven two-dimensional Navier-Stokes equations (Kolmogorov flow) is presented. Different regimes are studied, all corresponding to a chaotic behavior of the flow. The diffusion is found to be strongly asymmetric with a very weak transport perpendicular to the forcing direction. The trajectories of the particles are characterized by the presence of traps and flights. The trapping time distributions show algebraic decrease, and strong anomalous diffusion is observed in transient phases. Different regimes lead to different types of diffusion, i.e., no universal behavior of diffusion is observed, and both time and space properties are needed to define anomalous transport. (c) 2001 American Institute of Physics.     

Monte Carlo methods for turbulent tracers with long range and fractal random velocity fields

Monte Carlo methods for computing various statistical aspects of turbulent diffusion with long range correlated and even fractal random velocity fields are described here. A simple explicit exactly solvable model with complex regimes of scaling behavior including trapping, subdiffusion, and superdiffusion is utilized to compare and contrast the capabilities of conventional Monte Carlo procedures such as the Fourier method and the moving average method; explicit numerical examples are presented which demonstrate the poor convergence of these conventional methods in various regimes with long range velocity correlations. A new method for computing fractal random fields involving wavelets and random plane waves developed recently by two of the authors [J. Comput. Phys. 117, 146 (1995)] is applied to compute pair dispersion over many decades for systematic families of anisotropic fractal velocity fields with the Kolmogorov spectrum. The important associated preconstant for pair dispersion in the Richardson law in these anisotropic settings is compared with the one obtained over many decades recently by two of the authors [Phys. Fluids 8, 1052 (1996)] for an isotropic fractal field with the Kolmogorov spectrum. (c) 1997 American Institute of Physics.     

On developed hydrodynamic turbulence spectra

New results in the theory of the developed hydrodynamic turbulence spectra are reviewed. Within the limits of the hypothesis of interaction locality it is shown that the series of equations for the moments has a scale-invariant solution with the Kolmogorov index values. With the help of the Wyld diagram technique the equations in the Direct Interaction Approximation are formulated which accurately take into account the transfer effect and have the precise solution in the form of the Kolmogorov spectrum. In the framework of these equations the corrections to the Kolmogorov spectrum due to gyrotropy and compressibility are found.     

A heat transfer model for phase change thermal energy storage

The modelling of the heat-transfer process in a phase-change storage medium consisting of parallel flow channels is presented. The model describing the mechanism of heat transfer from the process air stream to the surface of phase-change material by convection and diffusion of heat by conduction in the media consists of coupled, non-linear partial differential equations. These equations are solved by a finite difference scheme. Numerical solutions are used to study the performance of phase-change storage media during a single blow operation. A parametric study is also carried out for different non-dimensional parameters and operating conditions.     

Non-Equilibrium Statistical Mechanics¶of Strongly Anharmonic Chains of Oscillators

We study the model of a strongly non-linear chain of particles coupled to two heat baths at different temperatures. Our main result is the existence and uniqueness of a stationary state at all temperatures. This result extends those of Eckmann, Pillet, Rey-Bellet [EPR99a, EPR99b] to potentials with essentially arbitrary growth at infinity. This extension is possible by introducing a stronger version of Hörmander's theorem for Kolmogorov equations to vector fields with polynomially bounded coefficients on unbounded domains.     

Non-Markov stochastic processes satisfying equations usually associated with a Markov process

There are non-Markov Ito processes that satisfy the Fokker-Planck, backward time Kolmogorov, and Chapman-Kolmogorov equations. These processes are non-Markov in that they may remember an initial condition formed at the start of the ensemble. Some may even admit 1-point densities that satisfy a nonlinear 1-point diffusion equation. However, these processes are linear, the Fokker-Planck equation for the conditional density (the 2-point density) is linear. The memory may be in the drift coefficient (representing a flow), in the diffusion coefficient, or in both. We illustrate the phenomena via exactly solvable examples. In the last section we show how such memory may appear in cooperative phenomena.     

The maximal randomness principle in d-dimensional turbulence

A set of self-consistent equations in one-loop approximation in a statistical model of fully developed homogeneous isotropic turbulence, which is based on the maximal randomness principle of the incompressible velocity field with stationary energy spectral flux, is obtained. Thanks to the applied principle the model statistics becomes essentially non Gaussian. The set of equations does not possess the infrared and ultraviolet divergences near the obtained Kolmogorov spectral exponents. The solution of these equations leads to the Kolmogorov exponents, but its amplitude proportional to the Kolmogorov constantC _k is negative for Euclidean dimensiond=3. Systematic investigation is made of (inertial) steady state scaling solutions for dimensions 2<d<2.55695,where constantC _k (d) becomes positive. Considered in this way, the model stability is discussed in the context of widely studied fractal aspects of turbulence.We have greatly benefited from discussions with Dr. Altaisky from JINR Dubna. The authors (M.H. and M.S.) are grateful to D. I. Kazakov and to director D. V. Shirkov for hospitality at the Laboratory of Theoretical Physics, JINR, Dubna.This work was supported by Fundamental Research Russian Fund, International Scientific Fund (grant R-63000) and by Slovak Grant Agency for Science (grant 2/550/93).     

A stochastic model for solidification

A 3-dimensional (2-space, 1-time) model relating the diffusion of heat and mass to the kinetic processes at the solid-liquid interface, using a stochastic approach is presented in this paper. This paper is divided in two parts. In the first part the basic set of equations describing solidification alongwith their analysis and solution are given. The process of solidification has a stochastic character and depends on the net probability of transfer of atoms from liquid to the solid phase. This has been modeled by a Markov process in which knowledge of the parameters at the initial time only is needed to evaluate the time evolution of the system. Solidification process is expressed in terms of four coupled equations, namely, the diffusion equations for heat and mass, the equations for concentration of the solid phase and for rate of growth of the solid-liquid interface. The position of the solid-liquid interface is represented with the help of a delta function and it is defined as the surface at which latent heat is evolved. A numerical method is used to solve the equations appearing in the model. In the second part the results i.e. the time evolution of the solid-liquid interface shape and its concentration, rate of growth and temperature are given.     

Fractional backward Kolmogorov equations

This paper derives the fractional backward Kolmogorov equations in fractal space-time based on the construction of a model for dynamic trajectories. It shows that for the type of fractional backward Kolmogorov equation in the fractal time whose coefficient functions are independent of time, its solution is equal to the transfer probability density function of the subordinated process X(Sα(t)), the subordinator Sα(t) is termed as the inverse-time α-stable subordinator and the process X(τ) satisfies the corresponding time homogeneous Ito stochastic differential equation.     

水中金属丝电爆炸拉氏磁流体动力学模拟方法

基于拉格朗日描述,建立了水中金属丝电爆炸的单温磁流体动力学模型,并给出一种高阶混合有限元离散求解方法。拉氏可压缩流体方程组中,速度定义在H1连续有限元空间,内能定义在L2间断有限元空间实现物质界面的精确捕捉,存在激波的区域引入张量人工粘性抑制数值振荡。磁扩散方程仅考虑周向磁通量密度,简化为标量方程,使用H1连续有限元方法离散求解。焦耳热和洛伦兹力作为源项引入实现磁流体方程的耦合。数值算例表明:磁扩散求解器能够求解存在不同电导率的多介质磁扩散问题;拉氏流体求解器能够精确追踪物质界面,具有较好的激波分辨能力;耦合RLC电路的磁流体求解器能够复现水中金属丝电爆炸加热相变、冲击波的产生与传播、放电模式转变等物理过程。     

光-力耦合受激布里渊散射方程组

 采用位移形式表征各向同性介质的运动，从而对传统的SBS耦合波方程组进行改造，使之适用于多维的情况并体现光学理论和力学理论的耦合。采用有限差分方法数值求解了光学-力学耦合型后向SBS方程组，得到了瞬态的位移场、速度场和应力场。计算结果表明，当泵浦光光强为2.0×10 ¹⁵W/m ²时，SBS可激发厚度为1cm的K9玻璃样品的表面产生出0.1nm量级的位移和10m/s量级的速度，对应的应力幅度达到10MPa量级。     

隐-显积分因子间断Galerkin方法求解二维辐射扩散方程

提出一种求解二维非平衡辐射扩散方程的数值方法.空间离散上采用加权间断Galerkin有限元方法,其中数值流量的构造采用一种新的加权平均;时间离散上采用隐-显积分因子方法,将扩散系数线性化,然后用积分因子方法求解间断Galerkin方法离散后的非线性常微分方程组.数值试验中在非结构网格上求解了多介质的辐射扩散方程.结果表明:对于强非线性和强耦合的非线性扩散方程组,该方法是一种非常有效的数值算法.     

Center-of-mass tomographic approach to quantum dynamics

In the tomography representation we propose a new approach, which describes the dynamics of quantum particles by the Kolmogorov equations for non-negative propagators. To solve the Kolmogorov equations we use a diffusive Markovian random processes described by the related nonlinear stochastic Langevin equations. As a result the dynamics of quantum particles is described by the proposed numerical scheme combining both Langevin dynamics and Monte Carlo methods. We test the developed approach by applying it to the wave packet dynamics in harmonic potentials and to particle tunneling through a barrier.     

A family of evolution equations with nonlinear diffusion,Verhulst growth,and global regulation: Exact time-dependent solutions

A family of evolution equations describing a power-law nonlinear diffusion process coupled with a local Verhulst-like growth dynamics, and incorporating a global regulation mechanism, is considered. These equations admit an interpretation in terms of population dynamics, and are related to the so-called conserved Fisher equation. Exact time-dependent solutions exhibiting a maximum nonextensive q$q$-entropy shape are obtained.     

NQR Spin Diffusion in an Inhomogeneous Internal Field

The theory of NQR spin diffusion is extended to the case of spin lattice relaxation and spin diffusion in an inhomogeneous field. Two coupled equations describing the mutual relaxation and the spin diffusion of the nuclear magnetization and dipolar energy were obtained by using the method of nonequilibrium state operator. The equations were solved for short and long times approximation corresponding to the direct and diffusion relaxation regimes.     

聚合物充填过程中裹气现象的数值模拟

针对聚合物充填过程中的裹气现象,采用一种有限元（FEM）-间断有限元（DG）耦合算法对其进行数值模拟。对于自由运动界面,采用水平集（Level Set）方法进行捕捉;用XPP（eXtended Pom-Pom）本构模型来描述黏弹性流体的流变行为。采用有限元-间断有限元耦合算法求解统一的流场方程,并采用隐式间断有限元求解XPP本构方程、Level Set及其重新初始化方程。数值结果与文献中的实验结果及模拟结果吻合较好,验证了数值方法的稳定性及准确性。分析带有非规则嵌件型腔内,注射速度及浇口尺寸对裹气现象的影响,裹气容易出现在较高注射速度及较小浇口的情形。     

直流道PEMFC两相流数学模型

建立直流道质子交换膜燃料电池(PEMFC)三维非等温两相流数学模型,基于质子交换膜与气体之间的水分传递特征,综合考虑电渗、浓度扩散及电化学反应作用的影响,发展了膜电极水分传递的非平衡扩散模型.并自主开发程序代码对电池内复杂的多物理场耦合传递过程进行数值模拟,研究PEMFC电极内气态水、液态水分布、质子膜含水量分布和水迁移特性等,分析单电池内部的温度分布特征,并获得电池极化性能曲线.