数值计算与计算机应用
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数值计算与计算机应用  2019, Vol. 40 Issue (3): 161-187    DOI:
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MARS:多相流界面追踪问题的理论框架及高阶数值方法
张庆海
浙江大学数学科学学院, 杭州 310058
MARS: THEORETICAL FRAMEWORK AND NUMERICAL ALGORITHMS FOR A NEW APPROACH OF INTERFACE TRACKING IN SIMULATING MULTIPHASE FLOWS
Zhang Qinghai
School of Mathematical Sciences, Zhejiang Univeristy, Hangzhou 310058, China
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摘要 界面追踪是多相流最基本最重要的子问题之一.现有方法的思路是把其中的几何和拓扑问题转化为求解数值偏微分方程,从而避免处理这些复杂的几何和拓扑结构.与此形成鲜明对比的是,我们提出的MARS理论和高阶数值方法试图运用几何和拓扑的工具来解决几何和拓扑的问题.这篇综述性文章将简明扼要的介绍MARS理论和其衍生方法的核心内容,包括殷空间(连续介质流相的数学模型)、殷空间上的布尔代数及其算法实现、流相拓扑变化的同调分析、捐献区间(标量守恒率下相空间中的粒子分类和通量计算解析解)、VOF方法的收敛阶证明、一个四阶精度的界面追踪方法cubic MARS、以及一个四阶及以上精度的曲率估计算法HFES.经典数值测试的结果表明cubic MARS和HFES无论在效率上还是精度上相对于现有方法都具有很大优势.
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关键词界面追踪   曲率估计   时空一致高阶   布尔代数   殷空间     
Abstract: Interface tracking (IT) is one of the most fundamental problems in the study of multiphase flows. In current IT methods, geometric and topological problems are avoided by converting them to numerically solving partial differential equations. In contrast, we tackle geometric and topological problems with tools in geometry and topology. This review paper is a brief summary of the MARS theory and associated numerical algorithms, including (1) the Yin space as a model of continua, (2) Boolean algebra on the Yin space, (3) homological analysis on topological changes of a flow phase, (4) the theory of donating regions for classifying fluxing points and calculating Lagrangian flux in the context of scalar conservation laws, (5) numerical analysis on the convergence rates of VOF methods, (6) the cubic MARS methods for fourth-order interface tracking, and (7) the HFES method for estimating curvature and normal vectors with fourth-and higher-order accuracy. Results of classical benchmark tests show that the cubic MARS method and the HFES method are advantageous over current IT methods in terms of both accuracy and efficiency.
Key wordsinterface tracking   curvature estimation   high-order accuracy both in time and in space   Boolean algebra   Yin space   
收稿日期: 2019-05-30; 出版日期: 2019-09-12
作者简介: 张庆海,本科及硕士均毕业于清华大学水利系,2008年博士毕业于Cornell University环境流体力学专业,之后在美国Lawrence Berkeley National Lab以及University of Utah做计算数学方向的博士后研究.2016年起任浙江大学数学科学学院教授.研究流体力学和多相流中的计算数学;主要方向为界面追踪方法、高阶有限体积方法和自由边界问题等.针对标量守恒律提出了一个通量计算的解析结果,该成果以单独作者的身份发表在SIAM Review上.提出了一整套显式边界追踪的数学分析框架,证明了VOF等现有方法的二阶精度,提出了一系列四阶精度的界面追踪方法以及曲率估算方法.在不可压Navier-Stokes方程的数值求解上,提出了一个时空一致四阶精度的投影方法.
引用本文:   
. MARS:多相流界面追踪问题的理论框架及高阶数值方法[J]. 数值计算与计算机应用, 2019, 40(3): 161-187.
. MARS: THEORETICAL FRAMEWORK AND NUMERICAL ALGORITHMS FOR A NEW APPROACH OF INTERFACE TRACKING IN SIMULATING MULTIPHASE FLOWS[J]. Journal on Numerical Methods and Computer Applicat, 2019, 40(3): 161-187.
 
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