High-Order Time Stepping for the Incompressible Navier--Stokes Equations

Abstract

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(τ h −2), where τ 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.

An operator-splitting scheme for the stream function–vorticity formulation of the unsteady Navier–Stokes equations

“…If ∥ · ∥ and (·, ·) are the norm and the inner product in L 2 (Ω), then multiplying Eq. (7) by ψ we easily obtain: ∥f (x, s)∥ds (11) holds. This stability estimate must be satisfied, in some discrete sense, by any stable scheme for approximation of the Eqs.…”

“…This procedure can be extended to 3D, however, in such case it will eventually require the solution of one parabolic and one elliptic vectorial problems. So, it may not be competitive to various projection or artificial compressibility methods in primitive variables (see [10,11] for examples of such schemes). However, the present approach has one major advantage as compared to these schemes which is that it conserves mass exactly point wise (not just in a discrete sense) because from the discrete stream function values it is quite straightforward to produce a velocity approximation that is divergence free point wise.…”

Versatile anisotropic mesh adaptation methodology applied to pure quantity of interest error estimator. Steady, laminar incompressible flow

“…Then, we impose that c 2 =Δ t , where Δ t is controlled with a Courant number of 10. With a fully implicit Euler scheme, the velocity and the pressure equation are uncoupled by restructuring the equations . MINI element is still used.…”

An analysis of the Robert–Asselin time filter for the correction of nonphysical acoustics in an artificial compression method

“…|p n + 1 − p n − 1 | grows as k → 0 but is reduced by the Robert-Asselin (RA) filter. Here, m is the slope of the line-of-best-fit This is a common simplification made in the analysis of artificial compression and pressure projection methods, for example, [10][11][12][13][14][15][16]. Section 3 also gives a (necessarily technical) sharpening of the stability of pressure.…”