Finite Difference Methods for the Stokes and Navier–Stokes Equations

Solving numerically the Navier–Stokes equations on parallel systems

“…. } that will converge to the solution CD of (7). Most of the methods usually suggest that the nth element in this sequence (the one that is calculated in the nth iteration level) is a linear combination of some k previous elements:…”

“…M and N are not functions of the iteration level n. Although this assumption may contradict the non-linear fashion of the operator L as defined by the NS equations, it holds at least locally in the iterative directions and therefore the conclusions may be applied only for a certain number of iterations (this approach is sometimes called the quasi-linear iterative approach). The main difference between the intermediate results of a substep and the one-step results of the iterative procedure in a given iteration level is that the substep's results do not have to be a solution to (7), not even in the convergent state, while the iteration procedure is designed to satisfy (7) in the convergent state. If N is a known stationary function (a .id thus should not depend on n), then in general it can be assumed that it depends only on R. For example,…”

Ellipticity, accuracy, and convergence of the discrete Navier-Stokes equations

“…As noted above, in the Armfield scheme and in many other schemes [2,3], the full ellipticity of the continuous equations is recovered by adding additional terms into the continuity equation. The terms introduced by Armfield, referred to as the elliptic correction terms in the remainder of this paper, are presented in the next section.…”

“…This behaviour is the result of the discrete system being nonelliptic at the grid scale wave number. Several methods have been suggested to recover the full ellipticity of the discrete system and prevent the occurrence of this oscillation [1][2][3][4][5]. In Armfield [ 1 ] a finite volume method was presented that was specifically designed to have an identical discrete ellipticity to the finite volume SIMPLE scheme defined on a staggered mesh [6][7][8][9], which in turn has an identical discrete ellipticity to the standard second-order central differencing for the Laplace operator.…”

Numerical solution of the steady stokes equations

“…the correct boundary condition for p ( see, e.g., discussion in Strikwerda (1984) ). Work using the original system (1) and (2) has also been done, but it still requires some additional conditions for p near the boundary ( Harlow and Welch (1965), Strikwerda (1984) ). Chorin (1968) proposed a projection method for the more general N avierStokes equations that determines p without any artificial conditions on p.…”