Efficient Numerical Methods for Viscous Incompressible Flows

本文针对非定常粘性不可压Navier-Stokes方程（NSE），提出了一类高效、精确的数值计算方法，该方法基于一种新的完全耗散的无约束NSE格式，而不是传统的Stokes算子仅在发散自由场中耗散的格式。这类NSE求解器是无条件稳定的，同时显式处理压力和对流项。而且，在这类有限元方法中，不需要所谓的inf-sup条件。对于一般的三维流体问题，NSE的求解成本大大降低到在每个时间步求解一个标准的热方程和一个标准的Poisson方程，这种方法的简单性也使PI能够发展一类复杂流体的数值方法，如磁流体力学、液晶聚合物、地球发电机、气候模拟和大涡湍流模拟;计算流体动力学已经从一种数学好奇心发展成为流体动力学的几乎每一个分支的重要工具，从航空航天推进到天气预报，自从数字计算机出现以来，它已经受到国际社会的广泛关注。 所提出的格式的精度和效率将使我们能够模拟一般的三维时间相关的流动与合理的周转时间。 预计提出的快速算法将成为一个重要的工具，为任何科学家和工程师在众多的科学和工业应用的当前利益。

The investigator develops a class of efficient and accurate numericalmethods for the unsteady viscous incompressible Navier-Stokesequations (NSE) based a new unconstrained formulation of NSE withfully dissipation in contrast to the traditional formulation where theStokes operator is dissipative only in the divergence freefields. This class of NSE solver is unconditionally stable withexplicit treatment of both pressure and convection terms. Moreover, inthis class of finite element methods, there is no requirement of theso called inf-sup condition. The cost of solving NSE is greatlyreduced to solving a standard heat equation and a standard Poissonequation at each time step for general three dimensional fluid problems.The simplicity of the method also enable the PI to develop a class ofnumerical methods for complex fluids such as magneto-hydrodynamics,liquid crystal polymers, geodynamo, climate modeling, and large eddyturbulence simulations;Computational Fluid Dynamics has grown from a mathematical curiosityto become an essential tool in almost every branch of fluid dynamics,from aerospace propulsion to weather prediction and has receivedextensive attention throughout the international community since theadvent of the digital computer. The accuracy and efficiency of theproposed schemes will allow us to simulate general three-dimensionaltime-dependent flows with a reasonable turn-over time. It is expectedthat the proposed fast algorithms will become an important tool formany scientists and engineers in numerous scientific and industrialapplications of current interest.