非定态中子迁移方程的离散纵标—多群逼近理论

本文对有界凸的非均匀介质中具各向异性散射和裂变的连续能量中子迁移的非定态方程,将方向和能量两个变量同时离散的所谓离散纵标——多群逼近方法建立起系统的数学理论,证明了: 1 非定态迁移方程的解,可由相应的非定态离散纵标——多群迁移系统的解逼近。 2 原迁移算子的占优本征值,可由离散纵标——多群迁移算子所确定的具非负本征函数且实部为最大的本征值逼近。 3 原迁移算子的占优本征值所相应的正本征函数,可由离散纵标——多群迁移算子的实部为最大的本征值所相应的非负本征函数逼近。 4 估计了各种逼近的阶。     

粒子输运方程的子网格平衡格式的稳定性和收敛性

本文研究四边形网格上求解粒子输运方程的有限体积格式,其中角方向变量采用离散纵标(Sn)方法,空间离散采用子网格平衡(SCB)格式.利用能量估计方法,证明了在正交网格上该格式的稳定性和离散解的收敛性.数值实验结果验证了格式的稳定性和离散解的收敛性.     

利用修正离散纵标法解决一类特殊迁移方程的特征值问题

利用修正的离散纵标法讨论平板几何各向异性散射的一类特殊的迁移方程特征值的求解问题,给出了这类迁移方程的具体形式及转化方法.最后通过Matlab编程算出近似值.     

关于平板几何反应堆临界中子通量离散纵标法的收敛速度

在核反应堆的理论和计算中,求临界中子通量是一个中心问题.它的存在性已在[3]中证明了,但是能够精确解出的仅只是极个别的情况.人们通常采用近似方法求解.经验说明,对一大类迁移问题采用离散纵标法是可以得到满意结果的,因而引起了人们研究离散纵标法的兴趣.本文的目的是讨论离散纵标法求解临界中子通量的收敛性和收敛速度.     

非定常线性化Navier-Stokes方程的非协调流线扩散有限元法分析

对非定常线性化Navier-Stokes方程提出了非协调流线扩散有限元方法．用向后Euler格式离散时间,用流线扩散法处理扩散项带来的非稳定性．速度采用不连续的分片线性逼近,压力采用分片常数逼近．得到了离散解的存在唯一性以及在一定范数意义下离散解的稳定性和误差估计．     

p≥1—阶拟总体列紧算子理论及其在线性迁移理论中的应用

在这篇文章中,我们在Banach空间中引进并研究了一类我们称之为p≥1—阶拟总体列紧算子,(p th-order quasi-collectively compact operator),这类算子最初的简单论述己在我们的报告[1]中提出,拟总体列紧算子类可视为是对1971年,由Anselone所引进和研究的那类总体列紧算子的一种特殊扰动的结果,将这类拟总体列紧算子理论应用于线性迁移问题,可建立起求解积分—微分Boltzmann方程某些近似方法,如离散纵标法(Discrete-Ordinaes Methods)的统一的理论基础。本文所论述的拟总体列紧算子逼近理论的应用,结合我们的工作,系统地给出了线性迁移理论中,高维离散纵标法的种种逼近,包括谱逼近的定性的理论阐述,从而回答了《第四届国际迁移理论会议》上所提出的有关离散纵标法的问题。     

具有非局部反应的时滞扩散Nicholson方程的行波解

对具有扩散项的时滞Nicholson方程的行波解进行了研究．特别是考虑到生物个体在空间位置上的迁移,研究了具有非局部反应的时滞扩散模型．对于弱生成时滞核,运用几何奇异摄动理论,在时滞充分小的情况下,证明了行波解的存在性．     

对流扩散方程的经济差分格式

１．引言 对流扩散方程是一类基本的运动方程,它可描述质量、热量的输运过程以及反应扩散过程等众多物理现象．寻找稳定、快速实用的数值方法,有着重要的理论和实际意义．标准的差分方法或有限元方法对它常常失效,根本原因在于“对流项”的存在．［１］提出了解对流扩散方程的特征线修正技术,这一方法考虑沿着特征线（流动方向）的离散,利用了对流扩散问题的物理力学性质,可以有效地克服数值振荡,保证数值解的稳定,尤其对“对流占优”的问题,这一方法有突出的优越性．这方面已有大量的理论和应用研究成果［２,３,７］．对大规模…     

离散纵标方法与广义解

§1 中子,γ光子及其它粒子输运方程和计算方法的研究是核工程、技术领域内的一个重要内容.对粒子输运方程各种问题解的理论与数值研究具有重要意义.对这方面的研究有许多工作.本文的目的是用离散纵标(DSN)方法来作二维粒子输运方程:     

反应扩散方程解的渐近性态

本文使用锥映象不动点指数的计算方法，讨论一类反应扩散方程正静态解的存在性，并给出方程的静态解渐适性态．然后，利用上，下解的方法讨论相应周期系统周期解的存在性及其渐近性态．     

A UNIFORM FIRST-ORDER METHOD FOR THE DISCRETE ORDINATE TRANSPORT EQUATION WITH INTERFACES IN X,Y-GEOMETRY

A uniformly first-order convergent numerical method for the discrete-ordinate transport equation in the rectangle geometry is proposed in this paper. Firstly we approximate the scattering coefficients and source terms by piecewise constants determined by their cell averages. Then for each cell, following the work of De Barros and Larsen [1, 19], the solution at the cell edge is approximated by its average along the edge. As a result, the solution of the system of equations for the cell edge averages in each cell can be obtained analytically. Finally, we piece together the numerical solution with the neighboring cells using the interface conditions. When there is no interface or boundary layer, this method is asymptotic-preserving, which implies that coarse meshes （meshes that do not resolve the mean free path） can be used to obtain good numerical approximations. Moreover, the uniform first-order convergence with respect to the mean free path is shown numerically and the rigorous proof is provided.     

On a domain decomposition for the transport equation: theory and finite element approximation

We describe a domain decomposition method applied to a boundaryvalue problem for a transport equation in two dimensions. Thisdecomposition leads to a family of problems coupled throughsuitable equations on the interfaces (Steklov-Poincar equations).Via sharp stability estimates, we prove the convergence of aniterative procedure that gives the solution of the Steklov-Poincar equation for the two-domain case. What precedes isdone both for the continuous problem and for its discretizationbased on a streamline diffusion finite element method.     

Numerical Simulation of Two-Phase Flow through Heterogeneous Porous Media

A mixed finite element method is combined to finite volume schemes on structured and unstructured grids for the approximation of the solution of incompressible flow in heterogeneous porous media. A series of numerical examples demonstrates the effectiveness of the methodology for a coupled system which includes an elliptic equation and a nonlinear degenerate diffusion–convection equation arising in modeling of flow and transport in porous media.     

Large-strain numerical solution for coupled self-weight consolidation and contaminant transport considering nonlinear compressibility and permeability

Based on the one-dimensional (1D) consolidation equation and advection-dispersion transport equation, this paper presents a large-strain numerical solution for coupled self-weight consolidation and contaminant transport in saturated deforming porous media considering nonlinear compressibility and permeability relationships. The finite difference method is used to solve the governing equations for consolidation and transport. The proposed numerical solution for consolidation accounts for vertical strain, soil self-weight, and nonlinearly changing compressibility and hydraulic conductivity during consolidation. The solution for solute transport accounts for advection, diffusion, mechanical dispersion, linear and nonlinear equilibrium sorption, and porosity-dependent effective diffusion coefficient. The proposed numerical solution is verified against a self-weight consolidation field tank test, an analytical solution in the literature, and the CST1 numerical model. Using the verified solution, a series of parametric study is conducted to investigate the effect of several important parameters on the contaminant transport process for confined disposal of dredged contaminated sediments. The results indicate that the consolidation process and contaminant transport process induced by soil self-weight- can be very different from those induced by the more traditional external surcharge loading. Treating the self-weight loading as traditional external surcharge loading can underestimate the rate of contaminant outflow, especially in the early times. The compressibility and permeability relationships of sediment and the type of loading (i.e., self-weight loading versus external surcharge loading) can all significantly affect the contaminant transport process for confined disposal of dredged contaminated sediment.     

Convergence of an operator splitting method on a bounded domain for a convection-diffusion-reaction system

We solve a convection-diffusion-sorption (reaction) system on a bounded domain with dominant convection using an operator splitting method. The model arises in contaminant transport in groundwater induced by a dual-well, or in controlled laboratory experiments. The operator splitting transforms the original problem to three subproblems: nonlinear convection, nonlinear diffusion, and a reaction problem, each with its own boundary conditions. The transport equation is solved by a Riemann solver, the diffusion one by a finite volume method, and the reaction equation by an approximation of an integral equation. This approach has proved to be very successful in solving the problem, but the convergence properties where not fully known. We show how the boundary conditions must be taken into account, and prove convergence in L1,loc of the fully discrete splitting procedure to the very weak solution of the original system based on compactness arguments via total variation estimates. Generally, this is the best convergence obtained for this type of approximation. The derivation indicates limitations of the approach, being able to consider only some types of boundary conditions. A sample numerical experiment of a problem with an analytical solution is given, showing the stated efficiency of the method.     

A numerical method for one-dimensional diffusion problem using Fourier transform and the B-spline Galerkin method

This paper deals with the diffusion transport equation in one dimensional space. The B-spline Galerkin approximation method together with the Fourier transform and Gauss-Hermite quadrature are proposed to compute approximate solution. We give some theoretical results such as existence and uniqueness of the solution and an upper error bound estimates. We also give some numerical examples to illustrate the effectiveness of a such method.     

Approximate analytical solution of two-dimensional space-time fractional diffusion equation

This work presents an iterative scheme for the numerical solution of the space-time fractional two-dimensional advection–reaction–diffusion equation applying homotopy perturbation with Laplace transform using Caputo fractional-order derivatives. The solution obtained is beneficial and significant to analyze the modeling of superdiffusive systems and subdiffusive system, anomalous diffusion, transport process in porous media. This iterative technique presents the combination of homotopy perturbation technique, and Laplace transforms with He's polynomials, which can further be applied to numerous linear/nonlinear two-dimensional fractional models to computes the approximate analytical solution. In the present method, the nonlinearity can be tackle by He's polynomials. The salient features of the present scientific work are the pictorial presentations of the approximate numerical solution of the two-dimensional fractional advection–reaction–diffusion equation for different particular cases of fractional order and showcasing of the damping effect of reaction terms on the nature of probability density function of the considered two-dimensional nonlinear mathematical models for various situations.     

Nitsche-XFEM for a transport problem in two-phase incompressible flows

We consider the transport of a dissolved species in a divergence-free immiscible incompressible two-phase flow modeled by a convection diffusion equation. The so-called Henry interface condition leads to a jump condition for the concentration at the interface between the two phases. This discontinuity of the solution render the numerical solution on unfitted meshes difficult. Furthermore time discretization on moving interfaces and handling typically convection dominant situations makes the overall problem delicate. We propose a numerical method using extended finite elements and a Nitsche-type technique combined with streamline diffusion stabilization. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An adaptive characteristics method for advective-diffusive transport

An adaptive characteristics method is presented for the solution of advective-diffusive groundwater transport problems. The method decomposes the transport processes into advective and diffusive transport components. Advective flows are defined by using a streamtube incrementing procedure, based on the method of characteristics, to define the position of advective front. Diffusive transport orthogonal to the front is represented by an array of propagating streamtube elements, with dimension determined from analytical solution of the one-dimensional diffusion equation. Adaptive time scaling is used to moderate the dimensions and aspect ratios of the advective and diffusive streamtube elements as appropriate to the dominant transport mechanism. Finite differences are used to solve for transport ahead of the advancing front. The distribution of streamtubes are predetermined from a direct boundary element algorithm. Comparison with analytical results for a one-dimensional transport geometry indicates agreement for Peclet numbers between zero and infinity. Solution for transport in two-dimensional domains illustrates excellent agreement for Peclet numbers from zero to 25.     

A compact finite difference scheme for solving a three‐dimensional heat transport equation in a thin film

Heat transport at the microscale is of vital importance in microtechnology applications. The heat transport equation differs from the traditional heat diffusion equation in having a second‐order derivative of temperature with respect to time and a third‐order mixed derivative of temperature with respect to space and time. In this study, we develop a high‐order compact finite difference scheme for the heat transport equation at the microscale. It is shown by the discrete Fourier analysis method that the scheme is unconditionally stable. Numerical results show that the solution is accurate. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 441–458, 2000