Inclusion of a term $-\gamma\nabla\nabla\cdot u$, forcing $\nabla\cdot u$ to be pointwise small, is an effective tool for improving mass conservation in discretizations of incompressible flows. However, the added grad-div term couples all velocity components, decreases sparsity and increases the condition number in the linear systems that must be solved every time step. To address these three issues various sparse grad-div regularizations and a modular grad-div method have been developed. We develop and analyze herein a synthesis of a fully decoupled, parallel sparse grad-div method of Guermond and Minev with the modular grad-div method. Let $G^{\ast}=-diag(\partial_{x}^{2},\partial_{y}^{2},\partial_{z}^{2})$ denote the diagonal of $G=-\nabla\nabla\cdot$, and $\alpha\geq0$ an adjustable parameter. The 2-step method considered is $$\begin{eqnarray} 1 &:&\frac{\widetilde{u}^{n+1}-u^{n}}{k}+u^{n}\cdot \nabla \widetilde{u}^{n+1}+\nabla p^{n+1}-\nu \Delta \widetilde{u}^{n+1}=f\text{ & }\nabla \cdot \widetilde{u}^{n+1}=0,\\ 2 &:&\left[ \frac{1}{k}I+(\gamma +\alpha )G^{\ast }\right] u^{n+1}=\frac{1}{k }\widetilde{u}^{n+1}+\left[ (\gamma +\alpha )G^{\ast }-\gamma G\right] u^{n}. \end{eqnarray}$$ We prove its unconditional, nonlinear, long time stability in $3d$ for $\alpha\geq0.5\gamma$. The analysis also establishes that the method controls the persistent size of $||\nabla\cdot u||$ in general and controls the transients in $||\nabla\cdot u||$ when $u(x,0)=0$ and $f(x,t)\neq0$ provided $\alpha>0.5\gamma$. Consistent numerical tests are presented.
翻译:包含 $- gamma\ nabla\ nabla\ cdot u $, 迫使 $- nabla\ cdot u mount small underformalizations. 然而, 添加的 grad- div 夫妇所有速度元件, 降低宽度, 增加每步必须解决的线性系统的条件数。 为解决这三个问题, 已经开发了各种稀有的 grad- div 正规化和模块化的 grad- div 方法。 我们在这里开发和分析一个完全脱coupled, 平行淡化的 Guermon 和 Minev 的升级- div 方法的合成 。 让 gasto- diag (\ party_ x% 2) 、\ lax 、\\\\\ lit\ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ; ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ; ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ 。 ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ 。 。 。