An efficient method for the incompressible Navier-Stokes equations on irregular domains with no-slip boundary conditions, high order up to the boundary
Massachusetts Institute of Technology（麻省理工学院）
David Shirokoff, Rodolfo Ruben Rosales
备注：
50 pages, 14 figures
发表信息：
Journal of Computational Physics, Volume 230, Issue 23, 20 September 2011, Pages 8619-8646
Common efficient schemes for the incompressible Navier-Stokes equations, such as projection or fractional step methods, have limited temporal accuracy as a result of matrix splitting errors, or introduce errors near the domain boundaries (which destroy uniform convergence to the solution). In this paper we recast the incompressible (constant density) Navier-Stokes equations (with the velocity prescribed at the boundary) as an equivalent system, for the primary variables velocity and pressure. We do this in the usual way away from the boundaries, by replacing the incompressibility condition on the velocity by a Poisson equation for the pressure. The key difference from the usual approaches occurs at the boundaries, where we use boundary conditions that unequivocally allow the pressure to be recovered from knowledge of the velocity at any fixed time. This avoids the common difficulty of an, apparently, over-determined Poisson problem. Since in this alternative formulation the pressure can be accurately and efficiently recovered from the velocity, the recast equations are ideal for numerical marching methods. The new system can be discretized using a variety of methods, in principle to any desired order of accuracy. In this work we illustrate the approach with a 2-D second order finite difference scheme on a Cartesian grid, and devise an algorithm to solve the equations on domains with curved (non-conforming) boundaries, including a case with a non-trivial topology (a circular obstruction inside the domain). This algorithm achieves second order accuracy (in L-infinity), for both the velocity and the pressure. The scheme has a natural extension to 3-D.
A Nitsche method for incompressible fluids with general dynamic boundary conditions
Charles University（查理大学）；Universität Konstanz（康斯坦茨大学）；Institute of Science and Technology Austria (ISTA)（奥地利科学技术研究所）；Technische Universität Darmstadt（达姆施塔特工业大学）
ROM for Viscous, Incompressible Flow in Polygons -- exponential $n$-width bounds and convergence rate
Weierstrass Institute for Applied Analysis and Stochastics (WIAS)（魏尔斯特拉斯应用分析与随机性研究所）；SISSA（的里雅斯特国际高等研究院）；Sant’Anna School of Advanced Studies（圣安娜高等学院）；ETH Zürich（苏黎世联邦理工学院）
H(curl)-based approximation of the Stokes problem with slip boundary conditions
Faculty of Mathematics, University of Duisburg-Essen（杜伊斯堡-埃森大学数学学院）；SAM, ETH Zürich（苏黎世联邦理工学院应用数学系）；Department of Mathematics, University of Vienna（维也纳大学数学系）
Stability and error analysis of IMEX-BDFk finite element schemes for the incompressible Navier-Stokes system
Beijing Jiaotong University（北京交通大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）
