Multigrid solution of the Euler equations with local preconditioning.

Abstract

A multigrid method for solution of the steady two-dimensional Euler equations is presented. The combination of local preconditioning with multigrid relaxation makes the multigrid method very efficient in obtaining steady-state solutions. The key to the success of this combination is the development of single-grid marching schemes with guaranteed high-frequency damping. An optimization formulation is described that may be used to obtain multi-stage schemes with superior damping; the optimization is taken over the high-frequency content in the Fourier footprint of the preconditioned spatial operator. Both standard and semi-coarsened multigrid have been considered, requiring optimization over different frequency domains. The optimization problem has been solved by the method of simulated annealing together with the downhill-simplex method. Tables of multi-stage coefficients have been presented that are based on the solution to this optimization problem. It is shown that the combination of local preconditioning and multi-stage time-stepping can produce relaxation schemes that boast strong high-frequency damping for the entire range of flow angles, Mach numbers, cell aspect-ratios and (for Navier-Stokes operators) cell-Reynolds numbers. Such schemes are ideally suited for use as relaxation schemes in a multigrid framework, particularly if semi-coarsening is used. In addition they are superior relaxation schemes if only a single grid is used, in comparison to other explicit marching schemes with or without local preconditioning. The preconditioning already accelerates the convergence to the steady state and the high-frequency damping provides robustness. Multigrid Euler solutions on structured meshes are presented as test cases. These numerical studies indicate that multigrid speed-ups of a factor of 3-4 may be obtained when local preconditioning is used. Studies also indicate that explicit residual-smoothing can further improve convergence rates by up to 25%, as well as improving robustness, with only a minimal increase in the computational effort required per update. The extension to Navier-Stokes operators and three space dimensions, and the implementation on unstructured meshes are also briefly considered.