Multigrid Methods II pp 272-284 | Cite as

# A comparison of **several** Mg-methods for the solution of the time-dependent Navier-Stokes equations

## Abstract

In this paper a comparative study of the two main multigrid concepts for the time-dependent compressible Navier-Stokes equations is presented. In the direct multigrid method the solution advances in time on all grid levels with the accuracy of the finest grid. This method can be used for explicit or implicit difference schemes. The indirect multigrid method is applied within an implicit difference scheme to accelerate the iteration process between two time levels. The explicit MacCormack scheme and the Switched Evolution/Relaxation scheme are used as relaxation schemes. All methods are second-order accurate in space. First, the methods are applied to the time-dependent compressible Couette flow problem. In this case the indirect method using flux-vector splitting for the convective terms and central approximations for the viscous terms of the Navier-Stokes equations shows the best rate of convergence. Compared to a single fine grid solution obtained with the explicit MacCormack scheme the multigrid solution reduces the computational work by more than a factor of ten. Secondly, the indirect multigrid method is used to determine complex subsonic and supersonic flows. In the subsonic case the flow past a flat plate for different Reynolds and Mach numbers 10^{3}⩽Re⩽10^{5}, 0.5⩽Ma⩽0.9 is computed; in the supersonic case the flow past a wedge for a Reynolds number Re=10^{4} and Mach numbers 1.5⩽Ma⩽3.0 is determined. In order to increase the rate of convergence second-order accurate flux-vector splitting with a flux limiter which takes into account the very different space steps has to be employed. The investigation shows that the number of time steps required to obtain a converged solution is nearly independent of the number of grid points for the subsonic case as well as for the supersonic case.

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