Citation Context

...0 (Ω) d . We assume that Γ is sufficiently smooth so that the following regularity properties hold (cf. [18]): (2.6) ∀v ∈ L 2 (Ω) d , �S(v)�2 + �∇r� ≤c�v�. The following properties of S are proved in =-=[8]-=-. Lemma 2.1. For all v in H 1 0 (Ω)d and all 0 <γ<1 we have In particular, (∇S(v), ∇v) ≥ (1 − γ)�v� 2 − c(γ)�v − v ⋆ � 2 , ∀v ⋆ ∈ H. (∇S(v), ∇v) =�v� 2 , ∀v ∈ V. Lemma 2.2. The bilinear form H −1 (Ω) ...

Citation Context

... and using (33) we find ∇p n = 1 ∆t (I − P)un − ν∆(I − P)u n = ν(∆P − P∆)u n = ν∇p n S . The fact that the scheme of Kim and Moin can be transformed into that of Timmerman et al. has been observed in =-=[GMS]-=-. Finally, if one starts from (26) and retains the linear damping term with λ = 1/∆t but omits the term ∇p n S that produces diffusive damping of the divergence at the continuous level, one arrives at...

Citation Context

...e the most straightforward to adapt. We have used an algorithm with an explicit time integration of the nonlinear terms and where the Stokes operator is inverted using a velocity correction technique =-=[17, 18]-=-. In our implementation, we adapt the ‘convective’ form (u·∇)u of the nonlinearity. Other variants of the nonlinear terms, such as ‘rotational’ or ‘skew-symmetric,’ could potentially be used as a star...

Citation Context

...ies: • A common strategy to decouple the computation of the pressure from velocity is to use a projection type scheme as in the case for Navier-Stokes equations (cf., for instance, a recent review in =-=[32]-=-). However, most of the numerical analysis for projection type schemes are limited to problems with constant density and viscosity. The few exceptions which deal with variable density areFebruary 9, ...

Citation Context

...tep pressure is evaluated and the momentum and velocity are corrected, as in projection methods for incompressible flows (see [5,36] for the original papers, [29] for a comprehensive introduction and =-=[19]-=- for a review of most variants). The Characteristic-Based Split (CBS) scheme (see [31] for a recent review or [44] for the seminal paper), developped in the finite-element context, belongs to this lat...

Citation Context

... of one-dimensional fourthorder equations using a Fourier expansion in all but one direction [54]. • Finally, for the general cases, we shall use a projection type scheme (see the recent review paper =-=[55]-=-) to decouple the computation of the velocity from the pressure. For example, we may use the new consistent splitting scheme introduced in [56]. To be specific, we assume here that the velocity is sub...

Citation Context

... The method uses a time splitting, solving separately the transport equation for the density (1.1) and the momentum equation (1.2), the constraint (1.3) being treated through a projection method, see =-=[GMS06]-=-. This is the methodology followed in [GQ00, FGQ01]. These contributions use the formulation (1.7) since a specific care is given to the kinetic energy relation. The (unconditional) stability of the s...