Journal of the Korean Society for Aeronautical & Space Sciences (한국항공우주학회지)

The Korean Society for Aeronautical and Space Sciences (한국항공우주학회)
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Convergence and Stability Analysis of LU Scheme on Unstructured Meshes: Part I - Euler Equations

비정렬 격자계에서 LU Implicit Scheme의 수렴성 및 안정성 해석 : Part I-오일러 방정식

  • 김주성 (한국과학기술원 항공우주공학과 대학원) ;
  • 권오준 (한국과학기술원 항공우주공학과)
  • Published : 2004.11.01
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Abstract

A comprehensive study has been made for the investigation of the convergence and stability characteristics of the LU scheme for solving the Euler equations on unstructured meshes. The von Neumann stability analysis technique was initially applied to a scalar model equation, and then the analysis was extended to the Euler equations. The results indicated that the convergence rate is governed by a specific combination of flow parameters. Based on this insight, it was shown that the LU scheme does not suffer any convergence deterioration at all grid aspect ratios, as long as the local time step is defined using an appropriate parameter combination.

본 연구에서는 비정렬 격자계에서 가장 많이 쓰이는 근사 해법 중의 하나인 LU 기법의 오일러 방정식에 대한 수렴성 및 안정성에 관한 연구를 수행하였다. 적절한 스칼라 모델 방정식을 사용하여 LU기법이 갖는 고유한 특성에 관해서 해석적으로 논의하였으며, 이를 system of equations 형태인 오일러 방정식으로 확장 해석하였다. 해석 결과 LU 기법의 수렴성 및 안정성은 격자 종횡비와 연관된 특별한 독립변수의 조합으로 표현되며, 이러한 독립 변수의 조합을 사용하여 어떠한 종횡비의 격자에 대해시도 수렴성 및 안정성의 저하 현상이 발생되지 않음을 보였다.

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References

  1. Weiss, J. M., Maruszewski, J. P., and Smith, W. A., "Implicit Solution of Preconditioned Navier-Stokes Equations Using Algebraic Multigrid," AIAA Journal, Vol. 37, No. 1, 1999, pp. 29-36. https://doi.org/10.2514/2.689
  2. Frink, N. T., "Assessment of an Unstructured Grid Method for Predicting 3-D Turbulent Viscous Flows," AIAA Paper 96-0292, 1996.
  3. Strang, W. Z., Tomaro, R. F., and Grismer, M. J., "The Defining Methods of Cobalt60: A Parallel, Implicit, Unstructured Euler/Navier-Stokes Flow Solver," AIAA Paper 99-0786, 1999.
  4. Anderson, W. K. and Bonhaus, D. L., "An Implicit Upwind Algorithm for Computing Turbulent Flows on Unstructured Grids," Computers & Fluids, Vol. 23, No. 1, 1994, pp. 1-21. https://doi.org/10.1016/0045-7930(94)90023-X
  5. Sharov, D. and Nakahashi, K., "Reordering of 3-D Hybrid Unstructured Grids for Vectorized LU-SGS Navier-Stokes Computations," AIAA Paper 97-2102, 1997.
  6. Venkatakrishnan, V., "c," ICASE Report, No. 95-28, 1995.
  7. Luo, H., Baum, J. D., and Lohner, R., "A Fast, Matrix-free Implicit Method for Compressible Flows on Unstructured Grids," J. Computational Physics, Vol. 146, 1998, pp. 664-690. https://doi.org/10.1006/jcph.1998.6076
  8. 강희정, 권오준, "비정렬 적응격자를 이용한 로터 정지 비행 공력 해석," 한국항공우주학회지, 제28권, 제8호, 2000, pp. 1-7.
  9. Ekici, K. and Lyrintzis, A. S., "Parallel Newton-Krylov Methods for Rotorcraft Aerodynamics," AIAA Paper 2001-2587, 2001.
  10. Jespersen, D. C., "Design and Implementation of a Multigrid Code for the Euler Equations," Applied Mathematics and Computation, Vol. 13, 1983, pp. 357-374. https://doi.org/10.1016/0096-3003(83)90020-6
  11. Buelow, P. E. O., Venkateswaran, S., and Merkle, C. L., "Grid Aspect Ratio Effects on the Convergence of Upwind Schemes," AIAA Paper 95-0565, 1995.
  12. Wright, M. J., Candler, G. V., and Prampolini, M., "Data-Parallel Lower-Upper Relaxation Method for the Navier-Stokes Equations," AIAA Journal, Vol. 34, No. 7, 1996, pp. 1371-1377. https://doi.org/10.2514/3.13242
  13. Whitfield, D. L. and Taylor, L. K., "Discretized Newton-relaxation Solution of High Resolution Flux-difference Split Schemes," AIAA Paper 91-1539, 1991.
  14. Buelow, P.E.O., Venkateswaran, S., and Merkle, C. L., "Stability and Convergence Analysis of Implicit Upwind Schemes," Computers & Fluids, Vol. 30, 2001, pp. 961-988. https://doi.org/10.1016/S0045-7930(01)00038-X
  15. Hirsch, C., "Numerical Computation of Internal and External Flows," John Wiley & Sons, 1988.
  16. Anderson, W. K., Thomas, J. L., and Whitfield, D. L., "Multigrid Acceleration of the Flux-Split Euler Equations," AIAA Journal, Vol. 26, No. 6, 1988, pp. 649-654. https://doi.org/10.2514/3.9949