Efficient Numerical Methods for Viscous Incompressible Flows

研究人员开发了一类针对非定常粘性不可压缩纳维-斯托克斯方程 (NSE) 的高效且准确的数值方法，该方法基于一种新的无约束 NSE 公式，具有完全耗散，这与斯托克斯算子仅在散度自由场中耗散的传统公式不同。此类 NSE 求解器通过对压力项和对流项的显式处理而实现无条件稳定。而且，在这类有限元方法中，不需要所谓的inf-sup条件。对于一般三维流体问题，求解 NSE 的成本大大降低为在每个时间步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，例如磁流体动力学、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几乎所有领域的必备工具。流体动力学的一个分支，从航空航天推进到天气预报，自数字计算机出现以来受到了国际社会的广泛关注。 所提出方案的准确性和效率将使我们能够以合理的周转时间模拟一般的三维时间相关流。 预计所提出的快速算法将成为许多科学家和工程师在当前感兴趣的众多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.