三维定常问题的高阶紧致差分格式向非定常问题的推广

将已经建立的求解三维定常对流扩散方程的高阶紧致差分格式直接推广到三维非定常对流扩散方程的数值求解,时间导数项利用二阶向后欧拉差分公式,所得到的高阶隐式紧致差分格式时间为二阶精度,空间为四阶精度,并且是无条件稳定的.数值实验结果验证了本文方法的精确性和稳健性.     

非定常对流扩散方程的高精度多重网格方法

由已有的求解定常对流扩散方程的高阶紧致差分格式出发,直接推导出了数值求解非定常对流扩散方程的一种高阶隐式紧致差分格式,其时间为二阶精度,空间为四阶精度,并且是无条件稳定的。为了加快传统迭代法在求解隐格式时在每一个时间步上的迭代收敛速度,采用了多重网格加速技术。数值实验结果验证了本文方法的高阶精度、高效性及高稳定性。     

求解对流扩散方程的紧致修正方法

提出了求解对流扩散方程的紧致修正方法,该方法是在低阶离散格式的源项中,引入紧致修正项,从而构造高阶紧致修正格式,并进行求解.采用紧致修正方法对典型的对流扩散方程进行计算.结果表明,紧致修正方法虽然与二阶经典差分方法建立在相同的结点数上,但紧致修正方法的精度与紧致方法的精度相同,均具有四阶精度.所以紧致修正方法可以在少网...     

构造定常对流扩散方程高精度紧致差分格式的新方法

以一维定常对流扩散方程的高精度差分格式为基础，提出了一种构造二维定常对扩散方程高精度紧致差分格式的新方法，并给出数值例子。     

反应扩散模型在图灵斑图中的应用及数值模拟

反应扩散方程模型常被用于描述生物学中斑图的形成.从反应扩散模型出发,理论推导得到GiererMeinhardt模型的斑图形成机理,解释了非线性常微分方程系统的稳定常数平衡态在加入扩散项后会发生失稳并产生图灵斑图的过程.通过计算该模型,得到图灵斑图产生的参数条件.数值方法中采用一类有效的高精度数值格式,即在空间离散条件下采用Chebyshev谱配置方法,在时间离散条件下采用紧致隐积分因子方法.该方法结合了谱方法和紧致隐积分因子方法的优点,具有精度高、稳定性好、存储量小等优点.数值模拟表明,在其他条件一定的情况下,系统控制参数κ取不同值对于斑图的产生具有重要的影响,数值结果验证了理论结果.     

水声射线传播的黎曼几何建模·基础理论

水声建模一般采用外嵌描述,即以欧氏空间固定坐标系等要素刻画水声信道.黎曼几何是弯曲空间上的内蕴几何学,更能反映流形的本质性质.水声学高斯波束模型借鉴自地震学,可有效避免传统射线追踪的弊端,在以Bellhop为代表的水声模型中得到广泛应用,是水声射线建模与应用的主流方法之一.传统水声射线建模的欧氏空间底流形假设,难以有效刻画高斯波束的弯曲特性.本文通过建立水声射线传播的黎曼几何基本理论,得到程函方程、动态射线方程及高斯波束模型的黎曼几何内蕴形式,分析了水声射线几何拓扑性质,指出水声射线模型中的焦散点等价于黎曼几何中的共轭点,高斯波束几何扩展是测地线沿雅可比场的偏离,波束声线会聚体现为声场正截面曲率作用下偏离的测地线在共轭点的交汇.为验证理论正确性与适用性,本文以水平分层距离相关环境为例,给出特定环境和坐标系下应用前序理论建模的具体方法.3个典型水声传播算例的仿真对比分析,表明水声传播黎曼几何理论模型是准确有效的,相比Bellhop模型所采用的计算方法,具有更为清晰的数学物理含义.本文基础理论可方便推广至曲面、三维各向异性等情形,为后续在三维弯曲球体流形、四维时变伪黎曼流形等声传播环境下的...     

反常扩散与分数阶对流-扩散方程

反常扩散现象在自然界和社会系统中广泛存在.考虑了扩散过程的时间相关和时空相关性，用非局域性的处理方法，在传统的二阶对流 扩散方程基础上，得到了分数阶对流 扩散方程，以此方程来描述反常扩散.在此方程中，弥散项和对时间的导数为分数阶导数所代替.由此分数阶对流 扩散方程，对传统的费克扩散定律进行推广，得到了广义的分数费克扩散定律，分数费克扩散定律说明某时刻空间中某点的流量不仅与其领域内的浓度梯度有关，而且与整个空间中其他不同点的粒子浓度、浓度变化的历史，甚至初始时刻的浓度有关.讨论了方程的解——分数稳定分布，并由此说明了扩散运动的平均平方位移是运移时间的非线性函数. 关键词： 扩散 分数阶微积分 稳定分布（Lévy分布） 费克扩散定律     

求解Navier-Stokes方程组的组合紧致迎风格式

给出一种新的至少有四阶精度的组合紧致迎风(CCU)格式,该格式有较高的逼近解率,利用该组合迎风格式,提出一种新的适合于在交错网格系统下求解Navier-Stokes方程组的高精度紧致差分投影算法.用组合紧致迎风格式离散对流项,粘性项、压力梯度项以及压力Poisson方程均采用四阶对称型紧致差分格式逼近,算法的整体精度不低于四阶.通过对Taylor涡列、对流占优扩散问题和双周期双剪切层流动问题的计算表明,该算法适合于对复杂流体流动问题的数值模拟.     

间断有限元方法求解一维非平衡辐射扩散方程

研究一维非平衡辐射扩散方程的数值方法.通过求解间断系数热传导方程的广义黎曼问题,得到一种带加权数值流量,基于该数值流量构造了一类新型的间断有限元方法.在时间离散上采用向后Euler方法,形成的非线性方程组采用Picard迭代求解.数值试验表明该方法具有捕捉大梯度的能力,而且能适应扩散系数间断的情形.     

化学反应体系中涨落的时间空间关联(Ⅰ) 涨落、扩散和波

本文着重分析了如何描述化学反应体系中密度涨落的空间关联,对照反应扩散方程,我们在密度涨落分布函数所满足的Master方程中引入了扩散项或压力项,对无限介质,解Master方程,得到了一系列波,第一支波是与反应扩散方程的解一致,描述了平均密度起伏在空间的传播,在气体中就是声波,其它各支波,则描述了局部密度涨落高阶矩的变化在空间的传播(平均密度不改变),其实就是局部机率分布函数畸变后的传播过程,我们称之为涨落波,文中还讨论了如何求解线性初值问题。     

EFFECT OF NUCLEAR VISCOSITY ON FISSION PROCESS

According to the fission diffusion model,the deformation motion of fission nucleues is regarded as a diffusion process of quasi-Brownian particles under fission potential,Through simulating such Brownian motion in two dimensional phase space by Monte-Carlo method,the effect of nuclear viscosity on Brownian particle diffusion is studied,Dynamical quantities,suchas fission rate,kinetic energy distribution on scission,and so on are numerally calculated for various viscosity coefficients,The results are reasonable in physics,This method can be easily extended to deal with multi-dimensional diffusion problems.     

Dynamical derivation of momentum diffusion coefficients at collisions of relativistic charged particles

An expression has been obtained for the diffusion tensor of particles in the momentum space on the basis of the dynamics of particles motion. The general equations have been used to determine the rms momentum spread at collisions of relativistic charged particles at times shorter than the time of randomization of particles motion and at greater times when motion is completely random.     

Influence of motion of radiating particles on the concentration distribution at the boundary of a flat,semi-infinite layer

The distribution of excited atoms near the surface limiting the half space has been determined. The transfer of the excited state by two mechanisms, absorption and reradiation of photons and space motion, has been taken into account. An analytical solution has been found for the integral equation describing joint diffusion of the photons and particles. The concentration of excited atoms at the half-space boundary, determined after accounting for their space motion, may differ by orders of magnitude from the values obtained by solving the Biberman-Holstein equation.     

Four-dimensional mapping model for two-frequency electron cyclotron resonance heating

Two-frequency electron cyclotron resonance heating (ECRH) is modelled by a four-dimensional symplectic mapping derived from the nonrelativistic single particle equations of motion. The model includes changes in parallel energy due to the spatially separate resonance zones, not given by previous two-dimensional models. Fixed points are located and their linear stability limits determined. Resonances in action space are calculated along with their widths and used to obtain the adiabatic barrier to heating. Quasilinear diffusion coefficients are derived for the stochastic regime and found to agree well with numerical calculations. The primary diffusion in perpendicular energy can couple to the parallel motion, leading to diffusion in parallel energy. The resulting diffusion coefficient is calculated analytically and compared with numerical results. The much weaker Arnold diffusion along a resonance layer is also treated analytically, yielding diffusion coefficients in reasonable agreement with numerical values.     

Particle transport in space-periodic potentials in underdamped systems

We consider the motion of an underdamped Brownian particle in a tilted periodic potentialin a wide temperature range. Based on the previous data and the new simulation results weshow that the underdamped motion of particles in space-periodic potentials can beconsidered as overdamped motion in the velocity space in the effective double-wellpotential. Simple analytic expressions for the particle mobility and diffusion coefficientare derived with the use of the presented model. These accurately match numericalsimulation results.     

Kinetic equation for the dilute Lorentz gas with attractive forces

The kinetic equation for the dilute Lorentz gas of particles with repulsive-attractive potential, is derived. For that purpose the distribution function is decomposed into two parts: the one corresponding to the bounded motion of the marked particle and the second corresponding to its unbounded motion; only the second part, as representing the diffusion, is considered.After the appropriate modification of the projection operator, a general kinetic equation for the diffusion part of the distribution function is obtained. The low density limit of the scattering operator of this equation is found. The regions of bounded motion are excluded in the integration over position space.For the square-well potential, the Laplace-transformed kinetic equation is also given in full detail.     

Laplace operator and random walk on one-dimensional nonhomogeneous lattice

A classical result of probability theory states that under suitable space and time renormalization, a random walk converges to Brownian motion. We prove an analogous result in the case of nonhomogeneous random walk on onedimensional lattice. Under suitable conditions on the nonhomogeneous medium, we prove convergence to Brownian motion and explicitly compute the diffusion coefficient. The proofs are based on the study of the spectrum of random matrices of increasing dimension.     

Diffusion kinetics of water molecules in a nanoporous adsorbent

The diffusion of water vapor into porous polymer adsorbent MN-200 has been investigated by diode laser spectroscopy with a time resolution of 0.1 s. The cascade character of diffusion, caused presumably by the fractality of the porous matrix space, is established by decomposition analysis. The collisionless (Knudsen) mode of water motion in pores, in which water molecules can be sorted according to their spin states, is selected.     

Stochastic modeling of experimental chaotic time series

Methods developed recently to obtain stochastic models of low-dimensional chaotic systems are tested in electronic circuit experiments. We demonstrate that reliable drift and diffusion coefficients can be obtained even when no excessive time scale separation occurs. Crisis induced intermittent motion can be described in terms of a stochastic model showing tunneling which is dominated by state space dependent diffusion. Analytical solutions of the corresponding Fokker-Planck equation are in excellent agreement with experimental data.     

Complimentary aspects of diffusion imaging and fMRI: II. Elucidating contributions to the fMRI signal with diffusion sensitization

Tissue water molecules reside in different biophysical compartments. For example, water molecules in the vasculature reside for variable periods of time within arteries, arterioles, capillaries, venuoles and veins, and may be within blood cells or blood plasma. Water molecules outside of the vasculature, in the extravascular space, reside, for a time, either within cells or within the interstitial space between cells. Within these different compartments, different types of microscopic motion that water molecules may experience have been identified and discussed. These range from Brownian diffusion to more coherent flow over the time scales relevant to functional magnetic resonance imaging (fMRI) experiments, on the order of several 10s of milliseconds. How these different types of motion are reflected in magnetic resonance imaging (MRI) methods developed for "diffusion" imaging studies has been an ongoing and active area of research. Here we briefly review the ideas that have developed regarding these motions within the context of modern "diffusion" imaging techniques and, in particular, how they have been accessed in attempts to further our understanding of the various contributions to the fMRI signal changes sought in studies of human brain activation.