when solving discretized PDE's with a multigrid relaxation strategy,
the ultimate goal is to provide a solution with $N$ unknowns in $O(N)$
operations. Since the multigrid strategy couples the grid data from
boundary to boundary, as an implicit scheme does, and the embedded
single-grid relaxation scheme only needs to remove high-frequency
error modes, an explicit single-grid scheme (point-implicit if there
are more unknowns per cell) should suffice. Nevertheless, achieving
scalable convergence with such an approach has proved very hard when
the equations are of mixed elliptic-hyperbolic type, such as the
steady compressible Euler equations.
In this lecture a fully explicit multigrid strategy for Euler
discretizations will be described, that does achieve the optimum
convergence on a structured grid. The accuracy of the spatial
discretization is not compromised, on the contrary: it actually
benefits from the numerical approach (for low Mach numbers).
The three crucial components to this strategy will be discussed :
- Local preconditioning , to make the system of equations
(Euler, Navier-Stokes) behave like a scalar equation. This, by itself,
is a powerful single-grid acceleration method, since it removes the
stiffness due to the spread in physical time scales.
- An optimally smoothing multistage single-grid scheme .
If the discretized equations are properly preconditioned, the spatial
eigenvalues are sufficiently clustered to allow efficient removal of
any combination of high-frequency modes, as required for semicoarsened
multigrid relaxation.
- Semi-coarsened multigrid relaxation . The semicoarsening
is needed to remove certain combinations of high- and low-frequency
modes that are not affected by the single-grid scheme. These arise
when the flow is aligned with the grid, or the cell-aspect ratio is
large, or the Mach number approaches unity.