A Remark on the Projection-3 Method
نویسنده
چکیده
We show that the continuous (in time) form of the projection-3 scheme proposed in [2] is not a proper approximation of the unsteady Navier-Stokes equations. Hence the projection-3 scheme and its variants are not appropriate for the numerical computation of the Navier-Stokes equations. The projection-3 scheme in [2] was proposed as a possible improvement over the projection-1 and projection-2 schemes in [2]. To better understand the nature of these projection schemes, we will first exploit an intrinsic relation between the three schemes in [2], [3]. The classical projection method for solving the unsteady Navier-Stokes equations ut − ν∆u + (u · ∇)u+∇p = f , divu = 0, ∈ Ω× R, (1) was initially proposed by Chorin [1] and Temam [8]. A semi-discretized version (named as projection-1 scheme in [2]) of the classical projection method applied to the Navier-Stokes equations with homogeneous Dirichlet boundary condition can be written as follows: let u = u0, we solve successively ũ n+1 and {u, p} by (ũ − u) ∆t − ν∆ũ + (ũ · ∇)ũ = f, in Ω, ũ|∂Ω = 0, (2) (u − ũ) ∆t +∇p = 0, divu = 0, in Ω, u · n|∂Ω = 0. (3) It is well known that the above scheme suffers from large splitting errors. A number of modified projection schemes have been proposed to improve the accuracy (see This work was partially supported by NSF grant DMS-9205300.
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