解二维时间相关光子扩散方程的交替方向隐式法

研究了应用交替方向隐式法求解非均匀生物组织中的二维时间相关光子扩散方程,并同解析解以及Monte Carlo模拟的结果作了比较,数值结果表明,该方法具有较好的稳定性和较高的精度.     

多体体系输运理论——反常扩散

主要介绍最近在多体体系输运理论的一些模型和动力学等工作, 特别是一些有关反常扩散方面的工作结果.     

对流扩散问题的高精度交替组分八点格式

给出对流扩散方程的一种高精度交替分组8点格式,可以用于并行计算,且无条件稳定.数值实验证实此格式具有高阶精度.     

带扩散项色散方程的交替分组差分方法

给出了逼近带扩散项色散方程的非对称差分格式,并用这些差分公式构造了求解色散方程的交替分段差分算法,算法是无条件稳定的,能直接在并行机上使用,并给出了模型问题的试验结果.     

求解Burgers方程的一种交替分段隐式算法

首先提出一个新的求解Burgers方程的差分格式,然后在此差分格式的基础上构造了便于并行计算的交替分段隐格式,并作了线性化稳定性分析.数值结果表明,本方法具有较高的精度,尤适于扩散项系数较小时的计算,且有效避免了数值结果的非物理振荡.     

求解扩散方程的一类交替分组四点方法

给出了一种交替分组四点格式,该方法绝对稳定,具有并行本性,便于在并行计算机上直接使用.并进行了数值试验.     

节块内嵌离散纵标(SN)方法的算法研究及程序开发

介绍节块内嵌离散纵标(S_N)方法求解三维堆芯中子输运/扩散方程的算法框架.在基于扩散理论的三维粗网节块展开方法(NEM)的算法体系中,用基于输运理论的径向二维细网节块离散纵标方法(NDOM)的内迭代过程,替代节块展开方法(NEM)内迭代的径向求解过程.该算法充分考虑了核电厂反应堆堆芯的三维结构特点,另一方面,也充分利用了已经成熟的三维粗网节块展开方法(NEM)和二维离散纵标方法(S_N)的研究成果,同时有效避免了利用离散纵标方法(S_N)求解三维中子输运方程所面临的计算内存和计算时间的瓶颈问题.编制开发二维多群节块离散纵标方法(NDOM)模块程序NS_NM和三维多群节块展开方法(NEM)模块程序MGNEM,并以此为基础编制开发节块内嵌S_N方法的模块程序HANWIND;其中,NS_NM为HANWIND求解两维问题的功能模块.针对OECD/NEA-2D C5G7MOX基准问题以及两环路核电厂三维堆芯的数值验算结果表明,节块内嵌S_N方法的算法开发及程序编制有效、切实可行.     

一种求解三维热辐射输运方程的整体预处理迭代方法及并行计算

为三维灰体热辐射输运方程的隐式离散纵标方法发展一个整体预处理迭代方法并研制并行程序.该方法采用组装线性代数方程组策略,同时求出所有离散方向上的辐射强度.借助预处理的Krylov子空间迭代法,避免复杂网格上扫描算法可能遇到的死锁问题,能够提高健壮性和计算效率.空间离散上采用一阶迎风有限体积格式.数值实验测试变形六面体网格...     

A second-order accurate numerical method for the two-dimensional fractional diffusion equation

Spatially fractional order diffusion equations are generalizations of classical diffusion equations which are used in modeling practical superdiffusive problems in fluid flow, finance and others. In this paper, we present an accurate and efficient numerical method to solve a fractional superdiffusive differential equation. This numerical method combines the alternating directions implicit (ADI) approach with a Crank–Nicolson discretization and a Richardson extrapolation to obtain an unconditionally stable second-order accurate finite difference method. The stability and the consistency of the method are established. Numerical solutions for an example super-diffusion equation with a known analytic solution are obtained and the behavior of the errors are analyzed to demonstrate the order of convergence of the method.     

惯性约束聚变内爆中基于多块结构网格的高效辐射扩散并行算法

针对三维球形靶丸内爆高效模拟需求和传统笛卡尔正交网格上辐射加源困难的问题, 发展一种多块结构非正交网格生成方法, 并基于此种计算网格提出高效的三维扩散格式并行算法, 将其应用于辐射流体方程组的求解和三维内爆不对称性的数值模拟, 数值结果显示了算法的有效性。并行性能测试显示该算法可扩展到5400个核上, 并行效率达到69%。     

Comparing Diffusion Approximation with Radiation Transfer Analysis for Light Transport in Tissues

The diffusion model that is an approximation of the equation of radiation transfer is typically used to describe photon migration in scattering-dominant media. In general biological tissue is highly scattering and very weakly absorbing against near-infrared light, yet it is heterogeneous and may contain relatively highly absorbing or low-scattering regions. Here applicability of the diffusion approximation over the radiative transfer theory for describing ultrafast laser transport in biological tissues is numerically studied and investigated over different kinds of tissue conditions and geometries. Tissues having tumors of different sizes, locations and nature as well as dual-tumor and low-scattering conditions are considered. Radiation transfer analysis is taken as a comparison objective and it is initially proved to be accurate in benchmark comparisons with Monte Carlo simulation. The results predict systematically about the compatible conditions where and when we can use the diffusion approximation and the conditions in which the diffusion approximation may provide misleading results.     

二维球坐标系中子输运方程的一种并行SN算法

针对二维球坐标系下中子输运方程的SN算法, 提出基于(单元, 方向)二元组的有向图模型, 在已有的基于有向图的并行流水线算法基础上, 设计粒度可控多级并行SN算法。其中, 采用区域分解和并行流水线相结合的方式挖掘空间-角度方向的并行度, 提出能群流水并行方法, 并通过设置合适的流水线粒度来平衡有向图调度、通信和空闲等待开销。实验结果表明: 该算法可以有效地求解二维球坐标系下的中子输运方程。在某国产并行机1920核上, 对于96万网格、60个方向、24能群、数十亿自由度的典型中子输运问题, 获得了71%的并行效率。     

New Models for the Solution of Intermediate Regimes in Transport Theory and Radiative Transfer: Existence Theory, Positivity, Asymptotic Analysis, and Approximations

In this paper we propose, derive, and establish the mathematical foundations of new models for the solution of intermediate regimes in transport theory and radiative transfer. These new models consist of coupling the transport equations with their diffusion approximations. Our mathematical theory includes the existence theory, the positivity of the solutions, and the asymptotic analysis. We also give the rate of the asymptotic decay. In order to solve the new coupled problem we propose to use the transmission time marching algorithm introduced and studied in refs. 10, 13–15. We then study the convergence of the resulting algorithm. These studies are based in an essential way on the methods we introduced in refs. 14, 15.     

A Pulsed Field Gradient Spin-Echo Method for Diffusion Measurements in the Presence of Internal Gradients

Over the past decade several pulsed field gradient stimulated-echo methods have been presented for diffusion measurements in heterogeneous media. These methods have reduced or eliminated the coupling between the applied magnetic field gradient and a constant internal magnetic field gradient caused by susceptibility changes throughout the sample. For many research purposes thez-storage delay between the second and third π/2 RF pulse has been included in order to increase the decay of the echo attenuation to an appropriate level and to increase the signal-to-noise ratio by avoidingT₂relaxation of the magnetization in parts of the pulse sequence. For these reasons a stimulated-echo method has been applied instead of a spin-echo method. When studying systems where it is necessary to keep the duration of the pulse sequence at a minimum, and one is not dependent on usingz-storage time to increase the echo attenuation or to study diffusion as a function of observation time, a spin-echo method should be chosen. Here we propose a bipolar pulsed field gradient spin-echo method which is well suited to this purpose, and preliminary diffusion measurements are presented as illustration.     

An efficient and robust numerical algorithm for estimating parameters in Turing systems

We present a new algorithm for estimating parameters in reaction–diffusion systems that display pattern formation via the mechanism of diffusion-driven instability. A Modified Discrete Optimal Control Algorithm (MDOCA) is illustrated with the Schnakenberg and Gierer–Meinhardt reaction–diffusion systems using PDE constrained optimization techniques. The MDOCA algorithm is a modification of a standard variable step gradient algorithm that yields a huge saving in computational cost. The results of numerical experiments demonstrate that the algorithm accurately estimated key parameters associated with stationary target functions generated from the models themselves. Furthermore, the robustness of the algorithm was verified by performing experiments with target functions perturbed with various levels of additive noise. The MDOCA algorithm could have important applications in the mathematical modeling of realistic Turing systems when experimental data are available.