A complete numerical analysis for finite volume methods to The Navier-Stokes equations

纳维-斯托克斯方程有限体积法的完整数值分析

• 批准号: 0813571
• 负责人: Xiu Ye
• 金额: --
• 依托单位: University of Arkansas Little Rock
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0813571&HistoricalAwards=false
• 关键词: complete; numerical; analysis; finite; volume

Like the finite element method, the finite volume method is a discretization technique for solving partial differential equations. Due to the local conservation property and other attractive properties such as robustness with unstructured meshes, the finite volume method is widely used in computational fluid dynamics. Unlike finite element methods, numerical analysis for the finite volume methods is very limited. Only handful of papers analyzing the finite volume approximation for the Navier-Stokes equations can be found. The goal of the proposed research is to establish a complete numerical analysis for a class of the finite volume methods to the Navier-Stokes equations, including establishing stability of the methods; deriving optimal priori error estimates, posteriori error estimates and superconvergence of the solutions; and proving convergence of adaptive procedures. Computational fluid dynamics covers wide range of problems from air flow around airplane to blood flow of human body. The mathematical foundations of any computational fluid dynamics problem are the Navier-Stokes equations. The finite volume method is the classical numerical method for the problem used most often in commercial software and research codes. The objective of the project is to provide mathematical theory for the finite volume method.

与有限元方法一样，有限体积法也是一种求解偏微分方程组的离散化技术。由于有限体积法具有局部守恒性和对非结构网格的稳健性等优点，因此在计算流体力学中得到了广泛的应用。与有限元方法不同，有限体积法的数值分析非常有限。分析N-S方程有限体积近似的论文屈指可数。该研究的目的是建立一类求解N-S方程的有限体积方法的完整的数值分析，包括建立方法的稳定性；推导解的最优先验误差估计、后验误差估计和超收敛；以及证明自适应程序的收敛。计算流体力学涵盖了从飞机周围的空气流动到人体血液流动的广泛问题。任何计算流体力学问题的数学基础都是纳维-斯托克斯方程。有限体积法是商业软件和研究代码中最常用的经典数值方法。该项目的目的是为有限体积法提供数学理论。