BDF2 standard and rotational incremental pressure-correction variants

Fix $d\ge 1$, $N\ge 2$, $h=1\mathrm{/}N$, $\nu >0$, $k>0$, and a step $n\ge 1$. Supply $u^{n-1},u^n\in V_h$ with zero discrete divergence, a mean-zero $p^n\in Q_h$, and $f^{n+1}\in V_h$ whose components each have zero discrete mean, $h^d{\sum }_{j\in J_N}{f}_{i,j}^{n+1}=0(1\le i\le d)\mathrm{.}$ Compute a predictor ${\tilde{u}}^{n+1}\in V_h$ from $\frac{3{\tilde{u}}^{n+1}-4u^n+u^{n-1}}{2k}+\nu A_h{\tilde{u}}^{n+1}+G_h{p}^n=f^{n+1},$ then solve for the unique mean-zero ${\varphi }^{n+1}\in Q_h$ from ${\mathrm{\Delta }}_h{\varphi }^{n+1}=\frac{3}{2k}{D}_h{\tilde{u}}^{n+1}\mathrm{.}$ Set $u^{n+1}={\tilde{u}}^{n+1}-(2k\mathrm{/}3)G_h{\varphi }^{n+1}$. Choose exactly one pressure update:

• Standard incremental: $p^{n+1}=p^n+{\varphi }^{n+1}$.
• Rotational incremental: $p^{n+1}=p^n+{\varphi }^{n+1}-\nu D_h{\tilde{u}}^{n+1}$.

In both branches the pressure is gauged in $Q_h$. Initial values $u^0,u^1,p^1$ are supplied by the caller; no initialization accuracy, stability, or convergence rate is asserted here. The separate well-definedness theorem Well-definedness of the periodic pressure-correction solves ↗ verifies the predictor and pressure solves for either branch.