High-Order Time Stepping for the Incompressible Navier-Stokes Equations
High-Order Time Stepping for the Incompressible Navier-Stokes Equations
复制标题
不可压缩纳维-斯托克斯方程的高阶时间步进
DOI：
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复制
发表时间：
2015
影响因子：
3.1
通讯作者：
P. Minev
中科院分区：
文献类型：
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作者：
J. Guermond;P. Minev
本文介绍了一种求解不可压Navier-Stokes方程的高阶时间步方法，它与耦合方法不同，不需要在每个时间步求解鞍点问题，也不像投影方法那样产生分裂误差和虚假边界层。该技术是人工压缩性方法的推广;它是无条件稳定的（对于非定常Stokes方程），可以在时间上达到任何阶，并且解耦速度和压力。与在每个时间步长要解决的全离散向量值问题相关联的线性系统的条件数类似于$O（\tau h^{-2}）$，其中$\tau$是时间步长，$h$是空间网格大小。没有泊松问题或其他二阶椭圆问题必须解决的压力修正。不像投影方法，最佳收敛观察数值Dirichlet和混合Dirichlet/Neumann边界条件。
This paper introduces a high-order time stepping technique for solving the incompressible Navier--Stokes equations which, unlike coupled techniques, does not require solving a saddle point problem at each time step and, unlike projection methods, does not produce splitting errors and spurious boundary layers. The technique is a generalization of the artificial compressibility method; it is unconditionally stable (for the unsteady Stokes equations), can reach any order in time, and uncouples the velocity and the pressure. The condition number of the linear systems associated with the fully discrete vector-valued problems to be solved at each time step scales like $O(\tau h^{-2})$, where $\tau$ is the time step and $h$ is the spatial grid size. No Poisson problem or other second-order elliptic problem has to be solved for the pressure corrections. Unlike projection methods, optimal convergence is observed numerically with Dirichlet and mixed Dirichlet/Neumann boundary conditions.