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Multigrid Solution of the Steady Euler Equations

  • P. W. Hemker
  • S. P. Spekreijse
Part of the Notes on Numerical Fluid Mechanics book series (NNFM, volume 11)

Summary

A multigrid (MG) method for the approximation of steady solutions to the full 2-D Euler equations is described. The space discretization is obtained by the finite volume technique and Osher’s approximate Riemann-solver. Symmetric Gauss-Seidel relaxation is applied to solve the nonlinear discrete system of equations. A multigrid method, the full approximation scheme, accelerates this iterative process.

In a few two-dimensional testproblems, (subsonic, transsonic and supersonic) the multigrid iteration is applied to an initial estimate that was obtained by means of the FMG-technique (nested iteration). For the discretization on the different levels, a fully consistent sequence of nested discretizations is used. The prolongations and restrictions selected are in agreement with this consistency.

It turns out that the total amount of work required to obtain a solution, that is accurate upto truncation error, corresponds to a small number of nonlinear Gauss-Seidel iterations. In the case of transsonic flow the rate of convergence of the MG-iteration appears independent of N, i.e. the number of cells in the discretization.

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References

  1. [1]
    Brandt, A., “Guide to Multigrid Development.” In: Multigrid Methods (W. Hackbusch and U. Trottenberg eds) Lect. Notes in Mathematics 960, pp. 220-312, Springer Verlag 1982.Google Scholar
  2. [2]
    Harten, A., Lax, P.D. & van Leer, B., “On upstream differencing and Godunov-type schemes for hyperbolic conservation laws.” SIAM Review 25 (1983) 35–61.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    Hemker, P.W. & Spekreijse, S.P., “Multiple Grid and Osher’s Scheme for the Efficient Solution of the Steady Euler Equations”. In preparation.Google Scholar
  4. [4]
    Jameson, A., “Numerical Solution of the Euler Equations for Compressible Inviscid Fluids.” In: Procs 6th International Conference on Computational Methods in Applied Science and Engineering, Versailles, France, Dec. 1983.Google Scholar
  5. [5]
    Jespersen, D.C., “Recent developments in multigrid methods for the steady Euler equations.” Lecture Notes, March 12–16, 1984, von Karman Inst., Rhode-St. Genese, Belgium.Google Scholar
  6. [6]
    Lax, P.D., “Hyperbolic systems of conservation laws and the mathematical theory of shock waves.” Regional conference series in applied mathematics 11. SIAM Publication, (1973).Google Scholar
  7. [7]
    Mulder, W.A. “Multigrid Relaxation for the Euler equations.” To appear in: J. Comp. Phys. 1985.Google Scholar
  8. [8]
    Osher, S. & Chakravarthy, S., “Upwind schemes and boundary conditions with applications to Euler equations in general geometries”. J. Comp. Phys. 50 (1983) 447–481.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    Osher, S & Solomon, F., “Upwind difference schemes for hyperbolic systems of conservation laws”. Math. Comp. 38 (1982) 339–374.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    Rizzi, A. & Viviand, H., (Eds) “Numerical Methods for the computation of inviscid transonic flows with shock waves.” Proceedings GAMM Workshop, Stockholm, 1979, Vieweg Verlag, 1981.Google Scholar
  11. [11]
    Roe, P.L., “Approximate Riemann solvers, parameter vectors and difference schemes.” J. Comp. Phys. 43 (1981) 357–372.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    Roe, P.L., “The use of the Riemann problem in finite difference schemes.” In: Procs. 7th Int. Conf. Num. Meth. Fl. Dyn. (1980), (Reynolds & McCormack eds.) Springer Lecture Notes in Physics 141, pp.354-359, Springer Verlag 1981.Google Scholar
  13. [13]
    Ron-Ho Ni, “A multiple grid scheme for solving the Euler equations.” AIAA Journal 20 (1982) 1565–1571.zbMATHCrossRefGoogle Scholar
  14. [14]
    Smoller, J., “Shock waves and reaction diffusion equations.” Grundlehren der mathematische Wissenschaften 258, Springer Verlag, 1983.Google Scholar
  15. [15]
    Steger, J.L., “A preliminary study of relaxation methods for the inviscid conservative gasdynamiscs equations using flux splitting.” Nasa Contractor Report 3415 (1981).Google Scholar
  16. [16]
    van Asselt, E.J., “On M-functions and nonlinear relaxation methods.” Report NW160/NW, Math. Centr., Amsterdam, 1983.Google Scholar
  17. [17]
    van Leer, B., “Flux-vector splitting for the Euler equations.” In: Procs. 8th Intern. Conf. on numerical methods in fluid dynamics, Aachen, June, 1982. Lecture Notes in Physics 170, Springer Verlag.Google Scholar
  18. [18]
    van Leer, B., “On the relation between the upwind-differencing schemes of Godunov, Engquist-Osher and Roe.” SIAM J.N.A. 5 (1984) 1.zbMATHGoogle Scholar

Copyright information

© Springer Fachmedien Wiesbaden 1985

Authors and Affiliations

  • P. W. Hemker
    • 1
  • S. P. Spekreijse
    • 1
  1. 1.Centre for Mathermatics and Computer ScienceCWIAmsterdamThe Netherlands

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