Error Analysis of Pressure-Correction Schemes for the Time-Dependent Stokes Equations with Open Boundary Conditions

Abstract

The incompressible Stokes equations with prescribed normal stress (open) boundary conditions on part of the boundary are considered. It is shown that the standard pressure-correction method is not suitable for approximating the Stokes equations with open boundary conditions, whereas the rotational pressure-correction method yields reasonably good error estimates. These results appear to be the first ever published for splitting schemes with open boundary conditions. Numerical results in agreement with the error estimates are presented.

An octree-based solver for the incompressible Navier–Stokes equations with enhanced stability and low dissipation

“…We shall demonstrate in the series of numerical experiments that the scheme retains second order accuracy if t varies smoothly. In the case of outflow boundary conditions, building a second order accurate stable pressure projection method is a well-known problem, see, e.g., [24,25]. It is not our intention to address this problem in the present paper.…”

“…It is not our intention to address this problem in the present paper. If one sets ν ∂ u n+1 ∂n Γ 2 = p n n in (10), then Guermond et al [24] proved that the splitting method is up to 3 2 order the accurate (the actual order depends on a certain regularity index). However, for such explicit treatment of pressure on outflow boundary, our experiments show instability if ν is not sufficiently large.…”

Efficient and scalable discretization of the Navier–Stokes equations with LPS modeling

“…In particular, we aim to compare in the framework proposed in this paper the rotational incremental pressure-correction scheme, cf. [42], to an alternative strategy proposed in [61], which has demonstrated to improve the accuracy for the standard incremental version while remaining compatible with the rotational one ;…”

Lagrange–Galerkin methods for the incompressible Navier-Stokes equations: a review

“…As we mention in the introductory section, LG methods yield every time step a linear Stokes problem, in the previous numerical tests we solve such Stokes problems by a direct method; however, Achdou and Guermond (2000) and Guermond and Minev (2003) apply LG methods in combination with fractional steps schemes to decouple velocity and pressure in a way that in the first step a velocity is calculated by solving a viscous equation satisfying the boundary conditions, then the pressure is obtained by solving a Poisson equation with homogeneous Neumann boundary conditions if the boundary conditions for the velocity are of Dirichlet type on Γ (see [14] for the case in which the velocity is also subject to open boundary conditions), and finally the divergence free velocity is calculated by a projection of the viscous velocity onto a divergence free subspace. Specifically, the projection/LG method of Guermond, Achdou and Minev to calculate a numerical solution to ((1))-( (3)) is the following.…”