# Numerical Study of Viscoplastic Flows Using a Multigrid Initialization Algorithm
## 2 Numerical Methodology
### Multigrid Procedure  
#### 2.5.1 Two-Grid Algorithm  
Iterative techniques are attractive because of their low storage overheads, specially for the solution of large systems of equations resulting from refined grids. In a CFD simulation, the efficiency of a solution technique is measured in particular by the computational cost put into achieving the desired accuracy. It is common knowledge that the discretization error is drastically reduced with highly refined meshes. However, the convergence rate towards the exact numerical solution of iterative methods, such as the Jacobi and Gauss-Seidel, rapidly gets worse as the mesh spacing is reduced. The application of the multigrid idea results in an approximately linear increase of computing time with grid refinement, allowing much finer grids to be used and therefore more accurate solutions to be obtained. In what follows, we provide an overview of FAS Multigrid concept applied to a two-grid system. A two grid algorithm consists of performing a few smoothing iterations on the fine grid, approximation of the required correction on the coarse grid, prolongation of the coarse grid correction to the fine grid and again smoothing on the fine grid. Consider a linear system of equations, whose exact solution for any variable on grid \(h\), \(\phi_{h}\), satisfies the following equation :  
\[A_{h}\phi_{h}=S_{h} \tag{27}\]  
Here, \(\phi\) is the exact solution at all CV centers of the grid with spacing \(h\), \(A\) is the coefficient matrix and \(S\) is the source term. After few SIMPLE iterations, the unconverged equation, having residual \(R_{h}\), can be expressed as:  
\[A_{h}^{*}\phi_{h}^{*}=S_{h}^{*}+R_{h} \tag{28}\]  
where \(A_{h}^{*}\) and \(S_{h}^{*}\) are evaluated using the approximate solution \(\phi_{h}^{*}\). Subtracting (27) from (28) yields  
\[A_{h}\phi_{h}=S_{h}+A_{h}^{*}\phi_{h}^{*}-S_{h}^{*}-R_{h} \tag{29}\]  
The obtained equation is used as the basis for multigrid coupling. The algebraic system (29) is then restricted on a coarser grid of spacing \(2h\), as follows:  
\[\widehat{A_{2h}\phi_{2h}}=\widehat{S_{2h}}+\widehat{A_{2h}[I_{h}^{2h}\widehat {\phi_{h}}]-\widetilde{S_{2h}}-[I_{h}^{2h}\widetilde{R_{h}}]} \tag{30}\]  
The restriction of variable values from fine \(h\) to coarse \(2h\) grid has to be performed by interpolation denoted by the operator \(I_{h}^{2h}\) as \(\widehat{\phi_{2h}}=I_{h}^{2h}\widehat{\phi_{h}}\). In this way, the coarse grid equations are derived. Variables and operators on grid \(2h\) based on the restricted approximate solution on the finer grid \(h\) are denoted by \((\sim)\), while those being modified in the course of iterations on the coarser grid \(2h\) are denoted by \((\wedge)\). The underlined terms are kept unchanged during the coarse grid iteration and can be considered as an extra source term in Equation (30). If the residual is zero \((\widetilde{R}=0)\), the above equation is satisfied, i.e., \((\widehat{\phi}=\widehat{\phi})\). For consistency reasons, \(\widehat{A_{2h}}\) and \(\widehat{S_{2h}}\) are evaluated on the coarse grid in the same way as \(\widehat{A_{2h}}\) and \(\widehat{S_{2h}}\) using the restricted values \([I_{h}^{2h}\widehat{\phi_{h}}]\). It is noted that the pressure operator is linear and, therefore, no restriction of pressure from the fine to coarse grid is needed. At each coarse grid iteration, the pressure is initialized as zero and the corrected pressure is then transferred to the fine grid to adjust the fine grid solutions. In contrast, the values \(\widehat{\phi_{2h}}\) will, in the course of coarse grid iterations, depart from their initial values \([I_{h}^{2h}\widehat{\phi_{h}}]\). After a certain number of SIMPLE iterations, the system (30) can be solved to obtain \(\widehat{\phi_{2h}}\).