Efficient Numerical Methods for Viscous Incompressible Flows

研究人员开发了一类高效和精确的非定常粘性不可压缩navier - stokes方程（NSE）的数值方法，该方法基于具有完全耗散的NSE的新的无约束公式，而不是传统公式中stokes算子仅在散度自由场中耗散。这类NSE解算器通过显式处理压力项和对流项是无条件稳定的。而且，在这类有限元方法中，不需要所谓的“内支撑条件”。求解NSE的成本大大降低到在一般三维流体问题的每个时间步上求解一个标准热方程和一个标准泊松方程。该方法的简单性也使PI能够开发一类复杂流体的数值方法，如磁流体动力学，液晶聚合物，地球发电机，气候建模和大涡流模拟；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几乎每一个分支的基本工具，从航空航天推进到天气预报，自从数字计算机出现以来，它已经在整个国际社会得到了广泛的关注。所提出方案的准确性和效率将使我们能够模拟具有合理周转时间的一般三维时间相关流。预计所提出的快速算法将成为许多科学家和工程师在许多当前感兴趣的科学和工业应用中的重要工具。

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.