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格子行走方法求解一维对流扩散方程

ISSN号：0469-5097
期刊名称：南京大学学报(自然科学版)
时间：2011.5.30
页码：276-280
分类：P641[天文地球—地质矿产勘探;天文地球—地质学]
作者机构：[1]南京大学地球科学与工程学院水科学系,南京210093
相关基金：国家自然科学基金（40725010,40672160）
相关项目：非均质条件下土壤—地下水系统中重非水相液体（DNAPLs）运移行为及其数值模拟研究

关键词：格子行走, 随机行走, 对流扩散方程, lattice-walk, random walk, convection-diffusion equation

中文摘要：

本文提出了全新的格子行走方法求解对流扩散方程．对流扩散方程广泛应用于水中污染物的运移模拟．然而在对流占优的情况下，常用的求解对流扩散方程的有限差分方法和有限元法会出现严重的数值弥散和数值振荡等不良数值现象．本文提出的格子行走方法，综合了随机行走方法（randomwalkmethod）和格点玻尔兹曼方法（1atticeBoltzmannmethod）的优点，有效克服了对流占优条件下的数值弥散现象．在给定格子长度和时间步长的情况下，格子行走方法所能计算的流速范围远远超过了有限差分方法的适用范围．格子行走方法构建了相邻时刻格点上溶质分布的转移矩阵（transitionmatrix），使得格点上的离散分布与高斯分布的0—2阶矩相等．通过转移矩阵，可以直接计算溶质分布随时间的变化过程．本文用一维瞬时点源模型和一维连续点源模型检验了格子行走方法的适用性．计算结果显示格子行走方法得到的结果与解析解吻合得非常好，而运用有限差分方法却出现严重的数值弥散．

英文摘要：

This paper presents a novel lattice-walk method to simulate the convection-diffusion equation. The convection diffusion equation is widely used to describe solute transport. However, conventional methods, e.g. the finite difference method or the finite element method, to solve this equation may result in numerical dispersion and oscillation, especially when convection velocity is large. The lattice-walk method combines the virtues of random walk method and lattice Boltzmann method （LBM）, to simulate convection-diffusion prbeess. The positions of solute particles are limited on lattice points and the jumping probability of particles on regular lattices is determined by Gaussian distribution of random walk. We construct the transition matrix by the equality of the moments （the zeroth, first and second order） of discrete distribution on By iteration, we can calculate the evolution of the concentration Then one-dimensional instantaneous point source model and constant concentration point source model are used to test this novel method. The results of one dimensional instantaneous point source model are shown. When convection dominates the convection-diffusion process, the result given by lattice-walk method agrees well with the analytical solution while the solution of finite difference method suffers from serious numerical dispersion. For the diffusion-dominated example, the concentration obtained from lattice-walk method is consistent with that from finite difference method. Therefore we can see the most important character of lattice walk method that it can simulate solute transport at convection velocities far beyond the range of conventional methods. The results of constant concentration point source model are shown. Our results of lattice-walk method agree well with analytical solutions and have little dispersion in the front of the concentration curve. However, the concentration obtained by finite difference method suffers rather serious numerical dispersion even if Pe number is not too large. There

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