Multigrid restriction

A method for smoothing the residual obtained on the fine grid in order to compute the corresponding residual on the coarse grid. In the terminology of the multigrid method, this step is called restriction. 

In one cycle of the multigrid method, first a few iterations are performed on the fine grid in order to obtain a comparatively smooth iteration error. After that the obtained residual is restricted to the coarse grid, where further iterations are performed in order to damp out the long-wave components of the solution error. Subsequently the coarse-grid solution is interpolated to the fine grid and the solution on the fine grid is updated. 

Recently, there has been strong interest in multigrid-type hybrid multiscale simulation. As depicted in Fig. 6, a coarse mesh is employed to advance the macroscopic, continuum variable over macroscopic length and time scales. At each node of the coarse mesh, a microscopic simulation is performed on a finer mesh in a simulation box that is much smaller than the coarse mesh discretization size. The microscopic simulation information is averaged (model reduction or restriction or contraction) to provide information to the coarser mesh by interpolation. On the other hand, the coarse mesh determines the macroscopic variable evolution that can be imposed as a constraint on microscopic simulations. Passing of information between the two meshes enables dynamic coupling. 

In practice, the illustrative two-stage procedure is replaced by more advanced multigrid cycles in which coarsening and refinement are used with special schedules of restriction and prolongation at different refinement levels. Common choices of multigrid cycles are the so-called V- and W-cycles. 

Figure 8 Standard multigrid structure for a V-, W-, and F-cycle. In this figure, the number of grid levels is four, the circles represent a Smoothing operation or an Exact solution, and the arrows indicate prolongation (upward) and restriction (downward) operations, flo represents the coarsest and 0,3 the finest grid.

There are basically two types of multigrid cycles V and W, as shown in Figure 6.16. The V cycle is more appropriate for solving supersonic and hypersonic flows, while the W cycle is used more for transonic and subsonic flows. For each loop some time steps (Runge-Kutta) are executed, variables are injected in the coarse grid, the residues are restricted, and the system is solved in the coarse grid. To return to the flne mesh, interpolate the corrections of coarse meshes to flne meshes. 

We continue in the ensuing chapters with several tutorials tied together by the theme of how to exploit and/or treat multiple length scales and multiple time scales in simulations. In Chapter 5 Thomas Beck introduces us to real-space and multigrid methods used in computational chemistry. Real-space methods are iterative numerical techniques for solving partial differential equations on grids in coordinate space. They are used because the physical responses from many chemical systems are restricted to localized domains in space. This is a situation that real-space methods can exploit because the iterative updates of the desired functions need information in only a small area near the updated point. 

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