A second-order backward difference time-stepping scheme for penalized Navier-Stokes equations modeling filtration through porous media

作者：Ravindran S S^*

来源：Numerical Methods for Partial Differential Equations, 2016, 32(2): 681-705.

DOI：10.1002/num.22029

• error estimates
• filtration thorough porous media
• linear extrapolation
• mixed finite element
• nonhomogeneous boundary condition
• penalty approach
• two-step backward difference

We investigate the stability and convergence of a fully implicit, linearly extrapolated second-order backward difference time-stepping scheme for the penalized Navier-Stokes equations modeling filtration through porous media. In the penalization approach, an extended Navier-Stokes equation is used in the entire computational domain with suitable resistance terms to mimic the presence of porous medium. It is widely used as an alternative to the heterogeneous approach in which different types of partial differential equations (PDEs) are used in fluid and porous subregions along with suitable continuity conditions at the interface. However, the introduction of extra resistance terms makes the penalized Navier-Stokes equations more nonlinear. We prove that the linearly extrapolated scheme is unconditionally stable and derive optimal order error estimates without any stability condition. To show feasibility and applicability of the approach, it is used to numerically solve a passive control problem in which flow around a solid body is controlled by adding porous layers on the surface.

• 出版日期2016-3