# Numerical Study of Viscoplastic Flows Using a Multigrid Initialization Algorithm
## 2 Numerical Methodology
### Multigrid Procedure
#### 2.5.1 Two-Grid Algorithm
Then, the coarse grid corrections for the full approximation velocity variables can be computed as:  
\[\delta\phi_{2h}=\widehat{\phi_{2h}}-I_{h}^{2h}\widehat{\phi_{h}} \tag{31}\]The corrections obtained on the coarsest grid are transferred back to the fine grid through the prolongation process denoted by \(l_{2h}^{h}\delta\phi_{2h}\), yielding a better estimate of the fine grid solution:  
\[\phi_{h}^{new}=\phi_{h}^{old}+\alpha_{MG}[l_{2h}^{h}\delta\phi_{2h}]=\phi_{h}^{ old}+\alpha_{MG}[l_{2h}^{h}(\widehat{\phi_{2h}}-l_{h}^{2h}\widehat{\phi_{h}})] \tag{32}\]  
where \(\alpha_{MG}\) represents the under-relaxation factor for coarse grid corrections. Then, few smoothing iterations are performed on the fine grid to eliminate any high-frequency error components introduced by the prolongation process. A two-grid correction scheme is carried out according to the following sequence of instructions:  
1. Initialize the set of primitive variables and impose boundary conditions;
2. At the finer multigrid level, execute few Simple iterations on the system (27);
3. Restrict the approximate solutions \(\phi_{h}^{*}\) and the correponding residuals \(R_{h}\);
4. Compute the coarse grid correction using Equation (30);
5. Check for convergence; if the solution is converged, prolongate back the velocities and pressures into the fine level;
6. The algorithm returns to Step 2 to perform smoothing iterations on the fine grid;
7. If the desirable residual reduction is not achieved yet, repeat Steps 2-6. Given (30), we assemble the equations for u-momentum and v-momentum on the coarse grid, whose solutions are \(\widehat{u}\), \(\widehat{v}\), as follows:  
\[\widehat{a_{p}}\widehat{a_{p}}=\sum_{nb}\widehat{a_{nb}}\widehat{a_{nb}}- \Delta y(p_{e}^{\prime}-p_{w}^{\prime})+\widehat{S_{u}}+\widehat{a_{p}} \widehat{a_{p}}\widehat{a_{p}}-\sum_{nb}\widehat{a_{nb}}\widehat{a_{p}}- \widehat{S_{u}}-\widehat{R_{u}} \tag{33}\]  
\[\widehat{a_{p}}\widehat{a_{p}}=\sum_{nb}\widehat{a_{nb}}\widehat{a_{nb}}- \Delta x(p_{n}^{\prime}-p_{s}^{\prime})+\widehat{S_{v}}+\widehat{a_{p}} \widehat{a_{p}}-\sum_{nb}\widehat{a_{nb}}\widehat{a_{p}}-\widehat{S_{v}}- \widehat{R_{v}} \tag{34}\]  
The SIMPLE procedure applied to the coarse grid equations involves the following steps:  
1. The primitive variables are initialized as: \[\widehat{u_{p}}=\widetilde{u_{p}},\ \ \widehat{v_{p}}=\widetilde{v_{p}},\ \ \ p^{\prime}=0\]
2. Few relaxation sweeps on the momentum equations are performed to obtain the approximate solutions \(\widehat{u_{p}^{*}}\) and \(\widehat{u_{p}^{*}}\). 3. The cell-face velocities are computed according to the momentum interpolation method. For the east interface of the volume cell, we can write: \[\widehat{u_{e}}=\left(\frac{\sum_{nb}a_{nb}\widehat{a_{nb}}+ \widehat{S_{u}}+\widetilde{f_{u}}}{a_{p}}\right)_{e}-\Delta y\frac{(p_{e}^{ \prime}-p_{p}^{\prime})}{(a_{p})_{e}}\] (35) \[A_{h}^{*}\phi_{h}^{*}=S_{h}^{*}+R_{h} \tag{28}\]0 (36) where \(\widetilde{f_{u}}\) and \(\widetilde{f_{v}}\) denote the underlying terms in (33) and (34), respectively. Analogous expressions for the other face velocities \(\widehat{u_{w}}\) and \(\widehat{v_{s}}\) can be derived in a similar manner. 4. The face velocities are substituted into the mass continuity equation to derive the coarse-grid pressure correction equation. The pressure correction equation is \(p^{\prime\prime}\), given by: \[A_{h}^{*}\phi_{h}^{*}=S_{h}^{*}+R_{h} \tag{28}\]1 (37) where \[A_{h}^{*}\phi_{h}^{*}=S_{h}^{*}+R_{h} \tag{28}\]2
5. Pressure equation (37) is solved, and the related pressure and velocity corrections are updated through \(p^{\prime\prime}\). 6. The algorithm returns to Step 2 to perform additional SIMPLE iterations. In practice, it is necessary to perform the above two-grid cycles continuously until the desired accuracy is achieved. The coarse grid problem (30) can be solved using an even coarser grid 4 \(h\) and so on. This whole process of going down from grid \(h\) to some coarsest grid and then back up again to grid h constitutes a multigrid cycle.