Bonos Series P

A geometric multigrid (GMG) solution procedure based on the box-smoother (Algorithm

4, Subsection 2.5.2.4) can be designed for the linearised finite volume discretisation of the Navier-Stokes equations as well. Before we continue to define the ingredients of this approach properly we emphasise that, in contrast to the finite element case from Subsection2.5.2.4, this approach does work reliably in a straightforward manner for the finite volume discretisation described in Section2.4.

A canonical way of defining the local degrees of freedom (local DOFs) for Algorithm

4 is to sweep over all mesh cells and for each cell to use all DOFs of the cell itself and all neighbouring cells, as depicted in Figure2.13. This results in local equation systems with 15 unknowns for each interior cell patch, as these consist of 5 cells with 3 unknowns each, see Figure 2.13. More complicated definitions are of course feasible as well, but unnecessary, as this simple approach works reasonably well.

The interpolation operator for the multigrid which is used in this work is defined analogously to the case of standard bilinear elements in FEM. That is, the interpolated value q(x) in child cell x is defined by the values in the parent cells a, b, c, d as

q(x) = 9 16q(a) + 3 16q(b) + 1 16q(c) + 3 16q(d),

see Figure2.14. The interpolation for the remaining child cells is defined by rotating the setup from Figure2.14accordingly. The restriction operator is defined as the transpose of

cell i

patch of corresponding cells

Figure 2.13: Cell patches used in the Box-smoother

d c

a _b

x

Figure 2.14: Cell centres a, b, c, d of parent cells to child cell x the interpolation operator.

This smoother has been tested as a multigrid preconditioner for a GMRES solver for the linearised systems (J + D)x = r from Algorithm 1. Table2.14 lists some results for this type of solution procedure for a driven cavity problem at different Reynolds num- bers Re and using different multigrid cycles as preconditioners. All results use a fixed time-step size of τ_k= 10. A fixed τk has been chosen to allow simple comparison of the

multigrid strategy for different Reynolds numbers and mesh refinement levels. The value chosen was found to form a reasonable compromise between computational expense for the solution of the linear systems and expense due to slow convergence to steady state at the higher end of the range of Reynolds numbers considered here. The linear systems are solved to a relative reduction to 10−3 of the preconditioned residual. This rather inac- curate solve is justified as finite time-steps are used, which limits the convergence of the nonlinear residual anyway. On the other hand even more inaccurate solves tend to hinder nonlinear convergence due to the non-uniform behaviour of the multigrid preconditioner. Tests for a range of problems and Reynolds numbers led to the choice of 10−3as stopping criterion as it proved a reasonable trade-off between robustness and computational cost.

current mesh is reduced to 10−8 of its initial norm. It is evident that for higher Re more time-steps are necessary. This reflects the increasingly nonlinear nature of the problem and longer time-scales for the diffusion of initial perturbations. Further, it becomes more pronounced on the finer meshes where artificial diffusion becomes small.

All tests listed in the Table2.14have been performed with damping parameter γ = 1.0, that is without damping. The first three columns of the table show results for the use of a simple V-cycle as preconditioner. While this performs well at low Re, its performance deteriorates significantly for higher Re. This deterioration may be attributed to the bad approximation of even the low frequency solution components on the coarse meshes due to the low order stabilisation terms introduced by the Roe scheme (2.4.11). Examples of the convergence of the solution of a lid driven cavity problem are given in Section

2.7.2.1where the Re = 1000 results demonstrate this effect. To verify that this is actually

the source of the deterioration the results of the simple V-cycle (a) (with one pre- and one post-smoothing step) are compared to those of a V-cycle with more smoothing (b) (one pre- and two post-smoothing steps) and a W-cycle (c) (pre-smoothing, coarse grid correction (CGC), post- and pre-smoothing, one more CGC, post-smoothing), see the last three columns in Table2.14.

The improved smoothing in (b) produces lower iteration counts for GMRES but is not sufficient to avoid deterioration as the mesh is refined. On the other hand the more expensive W-cycle (c) achieves this and results in what appear to be mesh independent iteration counts, compensating for the higher cost per cycle. This is illustrated by the total CPU time in each of the cases, which is the sum of the times taken on each level listed.

Note that the performance of the smoother for convection-dominated problems (high Re) may be improved by ordering the cells stream-wise, from inflow to outflow. However for the driven cavity problem it is difficult to define such an ordering due to the recircu- lation and the absence of in- and outflow boundaries. Therefore the results presented in Table2.14were produced with a simple coordinate based ordering which conforms with the flow direction at the boundary that drives the flow (the lid).

Overall we conclude that the box smoother used in a W-cycle as preconditioner for a GMRES solver provides a satisfying basis for a solver in the sense that the costs for the linear solves are linear in the number of unknowns and, in the Reynolds number range considered here, almost independent of Re.

Re level cells 10 100 1000 1000 1000 2 64 5×3 6×3 6×3 6×3 6×2.8 3 256 4×3 6×4 7×5 7×3.7 7×4.1 4 1024 4×3 6×4 8×6 8×4.8 8×4.8 5 4096 4×3 6×5.5 9×7.3 9×9 9×5.2 6 16384 4×4 6×6 11×10.5 11×8.5 11×6.1 7 65536 4×4.5 6×7 12×15.7 12×12.1 12×6.2 8 262144 6×5.5 6×7 21×21.1 14×15.1 14×5.1

(time-steps) × (average GMRES iterations)

total CPU time 32m52s 42m13s 197m38s 228m03s 118m41s

final residual 6.1e-09 1.4e-10 9.6e-11 9.5e-11 9.5e-11

cycle a a a b c

cycle types:

a V-cycle down, CGC, up b V-cycle down, CGC, up, down c W-cycle 2×( down, CGC, up) smoother:

up box smoother, cell ordering forward down box smoother, cell ordering backward Table 2.14: Iteration counts for the FV solver