Advertisement

Second order accurate upwind solutions of the 2D steady Euler equations by the use of a defect correction method

  • S. P. Spekreijse
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1228)

Abstract

In this paper a description is given of first and second order finite volume upwind schemes for the 2D steady Euler equations in generalized coordinates. These discretizations are obtained by projection-evolution stages, as suggested by Van Leer. The first order schemes can be solved efficiently by multigrid methods. Second order approximations are obtained by a defect correction method. In order to maintain monotone solutions, a limiter is introduced for the defect correction method.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Anderson, W.T., Thomas, J.L., and Van Leer, B., "A comparison of finite volume flux vector splittings for the Euler equations" AIAA Paper No. 850122.Google Scholar
  2. [2]
    Böhmer, K., Hemker, P. & Stetter, H., "The Defect Correction Approach." Computing Suppl. 5 (1984) 1–32.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Chakravarthy, S.R. and Osher, S., "High resolution applications of the Osher upwind scheme for the Euler equations." AIAA Paper 83-1943,Proc.AIAA Sixth Computational Fluid Dynamics Conf.(Danvers,Mass.July 1983), 1983,pp363–372.Google Scholar
  4. [4]
    Chakravarthy, S.R. and Osher, S., "A new class of high accuracy TVD schemes for hyperbolic conservation laws." AIAA Paper 85-0363,AIAA 23rd Aerospace Science Meeting. (Jan.14–17, 1985/Reno,Nevada).Google Scholar
  5. [5]
    Godunov, S.K., "A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics." Mat.Sb.(N.S.)47(1959),271-;also Cornell Aeronautical Laboratory transl..Google Scholar
  6. [6]
    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.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Hemker, P.W., "Defect correction and higher order schemes for the multi grid solution of the steady Euler equations." In this volume.Google Scholar
  8. [8]
    Hemker, P.W. & Spekreijse, S.P., "Multigrid solution of the Steady Euler Equations." In: Advances in Multi-Grid Methods (D. Braess, W. Hackbusch and U. Trottenberg eds) Proceedings Oberwolfach Meeting, Dec. 1984, Notes on Numerical Fluid Dynamics, Vol.11, Vieweg, Braunschweig, 1985.Google Scholar
  9. [9]
    Hemker, P.W. & Spekreijse, S.P., "Multiple Grid and Osher's Scheme for the Efficient Solution of the the Steady Euler Equations." Report NM-8507, CWI, Amsterdam, 1985.zbMATHGoogle Scholar
  10. [10]
    Mulder, W.A. "Multigrid Relaxation for the Euler equations." To appear in: J. Comp. Phys. 1985.Google Scholar
  11. [11]
    Osher, S. & Solomon, F., "Upwind difference schemes for hyperbolic systems of conservation laws." Math. Comp. 38 (1982) 339–374.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Roe, P.L., "Approximate Riemann solvers, parameter vectors and difference schemes." J. Comp. Phys. 43 (1981) 357–372.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Steger, J.L. & Warming, R.F., "Flux vector splitting of the inviscid gasdynamics equations with applications to finite difference methods." J. Comp. Phys. 40 (1981) 263–293.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Sweby, P.K. "High resolution schemes using flux limiters for hyperbolic conservation laws", SIAM J.Numer.Anal. 21 (1984) 995–1011.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    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
  16. [16]
    Van Leer, B., "Upwind difference methods for aerodynamic problems governed by the Euler equations." Report 84-23, Dept. Math. & Inf., Delft Univ. Techn., 1984.Google Scholar
  17. [17]
    Van Leer, B., "Towards the ultimate conservative difference scheme.2. Monotonicity and conservation combined in a second order scheme." J.Comp.Phys. 14,361–370(1974).CrossRefzbMATHGoogle Scholar
  18. [18]
    Van Leer, B. & Mulder, W.A., "Relaxation methods for hyperbolic equations." Report 84-20, Dept. Math. & Inf., Delft Univ. Techn., 1984.Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • S. P. Spekreijse
    • 1
  1. 1.CWI, Centre for Mathematics and Computer ScienceAmsterdamThe Netherlands

Personalised recommendations