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

• W. Schröder
• D. Hänel
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1228)

## 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 103⩽Re⩽105, 0.5⩽Ma⩽0.9 is computed; in the supersonic case the flow past a wedge for a Reynolds number Re=104 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.

## References

1. [1]
Brandt, A.: Guide to multigrid development. In "Lecture Notes in Mathematics" Vol. 960, pp. 220–312, Springer Verlag Berlin, 1981.Google Scholar
2. [2]
Ni, R.H.: A Multiple-Grid Scheme for Solving the Euler Equations, AIAA Paper No. 81-1025, 1981.Google Scholar
3. [3]
Schmidt, W., Jameson, A.: Applications of the Multi-grid Methods for Transonic Flow Calculations. In "Lecture Notes in Mathematics", Vol. 960, pp. 599–613, Springer Verlag Berlin, 1981.Google Scholar
4. [4]
Chima, R.V., Johnson, G.M.: Efficient Solution of the Euler and Navier-Stokes Equations with a Vectorized Multiple Grid AlgorithmGoogle Scholar
5. [5]
Jameson, A.: Solution of the Euler Equations for Two Dimensional Transonic Flow by a Multigrid Method, presented at the International Multigrid Conference, Copper Mountain, 1983.Google Scholar
6. [6]
Mulder, W.A.: Multigrid Relaxation for the Euler Equations. In "Lecture Notes in Physics", Vol. 218, pp. 417–421, Springer Verlag Berlin, 1985
7. [7]
van Leer, B.: Flux-vector splitting for the Euler equations. In "Lecture Notes in Physics", Vol. 170, pp. 507–512, Springer Verlag Berlin, 1982.Google Scholar
8. [8]
van Leer, B., Mulder, W.A.: Relaxation Methods for Hyperbolic Equations. In Proceedings of the INRIA Workshop on Numerical Methods for the Euler Equations for Compressible Fluids", Le Chesnay, France, Dec. 1983; to be published by SIAM.Google Scholar
9. [9]
MacCormack, R.W.: The effect of viscosity on hypervelocity impact cratering. AIAA Paper No. 69-354, 1969.Google Scholar
10. [10]
van Leer, B.: Towards the Ultimate Conservative Difference Scheme V. A Second-Order Sequel to Godunov's Method, J. Comp. Phys. 32 (1979), 101–136.
11. [11]
van Albada, G.D., van Leer, B., Roberts, W.W.: A Comparative Study of Computational Methods in Cosmic Gas Dynamics, Astron. Astrophys. 108, 1982, pp. 76–84.
12. [12]
Beam, R.M., Warming, R.F.: An implicit factored scheme for the compressible Navier-Stokes equations. Proceedings of the AIAA 3rd. Computational Fluid Dynamics Conference, Albuquerque, New Mexiko, 1977.Google Scholar
13. [13]
Mulder, W.A., van Leer, B.: Implicit Upwind Methods for the Euler Equations. AIAA Paper No. 83-1930, 1983.Google Scholar