KSCE Journal of Civil and Environmental Engineering Research (대한토목학회논문집)

Korean Society of Civil Engeneers (대한토목학회)
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Numerical Simulation of Shallow Water Flow Using Multi-dimensional Limiting Process (MLP)

MLP기법을 적용한 천수흐름의 수치모의

  • 안현욱 (국가수리과학연구소 계산수리과학연구부) ;
  • 유순영 (국가수리과학연구소 계산수리과학연구부)
  • Received : 2011.09.26
  • Accepted : 2012.03.05
  • Published : 2012.04.30
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Abstract

MLP (Multi dimensional Limiting Process) is implemented to simulate shallow water flows, and its performance over conventional TVD limiters in multidimensional flows is verified through several numerical simulations. MLP was developed to control oscillations for multi-dimensional compressible flows and proved to improve accuracy, efficiency and robustness in compressible flows. In this study, we applies MLP to modeling shallow water equations(SWEs) given that the SWEs are amenable to be solved using the large range of numerical methods developed to deal with compressible flows and MLP has been yet used for SWEs. Simulation results through the benchmark tests show that MLP has favorable features such as numerical oscillation control and convergence behaviors comparable to the conventional limiters. Both numerical accuracy and stability are improved in multi-dimensional discontinuous flows.

천수방정식의 수치모형에 MLP(Multi dimensional Limiting Process)기법을 적용한 후 수치모의를 통해 MLP의 수치 진동 제어 성능을 검증하였다. MLP기법은 2, 3차원에서 기존의 TVD 제어자(limiter)들보다 안정적이며 정확한 수치모의를 가능하게 한다. 다차원에서 정확하고 안정적인 수치모의가 가능하도록 개발된 MLP기법은 압축성 유체를 표현하는 2, 3차원 오일러 방정식에 적용되어 기존의 제어자들에 비해 그 뛰어난 성능이 검증된 바 있다. 하지만 천수방정식에 적용된 예는 없으며, 이에 본 연구는 천수방정식에 MLP를 적용하고 천수방정식 수치모형 검증에 주로 사용되는 수치모의를 통해 MLP의 진동 제어 성능을 검증하였다. 모의 결과, MLP는 2차원 천수방정식에 있어서도 기존의 제어자들과 비교하여 수치진동을 보다 잘 제어하는 것으로 판단된다. MLP 사용으로 인해 불연속면 근처에서 정확도가 향상되었고 수치진동이 발생하지 않아 보다 안정적인 모의가 하게 되었다.

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References

  1. 강민구, 박승우(2003) ENO 기법을 이용한 2차원 유한체적 수치 모형, 한국수자원학회논문집, 한국수자원학회, 제36권, 제1호, pp. 1-11.
  2. 김병현, 한건연, 김지성(2009) Unsplit 기법을 적용한 흐름율과 생성항의 처리기법, 한국수자원학회논문집, 한국수자원학회, 제 42권, 제12호, pp. 1079-1079.
  3. 안현욱, 유순영(2011) 적응적 메쉬세분화기법과 분할격자기법을 이용한 극한 도시홍수 실험 모의. 한국수자원학회논문집, 한국수자원학회, 제44권, 제7호, pp. 511-522.
  4. An, H. and Yu, S. (2012) Well-balanced shallow water flow simulation on quadtree cut cell grids. Advances in Water Resources, Vol. 39, pp. 60-70. https://doi.org/10.1016/j.advwatres.2012.01.003
  5. Audusse, E., Bouchut, F., Bristeau, M.O., Klein, R., and Perthame, B. (2004) A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM Journal on Scientific Computing, Vol. 25(6), pp. 2050-2065. https://doi.org/10.1137/S1064827503431090
  6. Caleffi, V., Valiani, A., and Bernini, A. (2006) Fourth-order balanced source term treatment in central WENO schemes for shallow water equations Journal of Computational Physics, Vol. 218, pp. 228-245. https://doi.org/10.1016/j.jcp.2006.02.001
  7. Causon, D.M., Ingram, D.M., Mingham, C.G., Yang, G., and Pearson, R.V. (2000) Calculation of shallow water flows using a Cartesian cut cell approach. Advances in Water Resources, Vol. 23, pp. 545-562. https://doi.org/10.1016/S0309-1708(99)00036-6
  8. Chow, V.T. (1959) Open-channel hydraulics. McGraw-Hillm, New York.
  9. Goutal, N. and Maurel, F. (1997) Proceedings of the 2nd workshop on dam-break simulation. Note Technique EDF, HE-43/97/016/B.
  10. Harten, A. (1983) High resolution schemes for hyperbolic conservation laws, Journal of Computational Physics, Vol. 49, pp. 357-393. https://doi.org/10.1016/0021-9991(83)90136-5
  11. Kim, K. and Kim, C. (2005) Accurate, efficient and monotonic numerical methods for multi-dimensional compressible flows Part II: Multi-dimensional limiting process. Journal of Computational Physics, Vol. 208, pp. 570-615. https://doi.org/10.1016/j.jcp.2005.02.022
  12. Liang, Q., Borthwick, A.G.L., and Stelling. G. (2004) Simulation of dam- and dyke-break hydrodynamics on dynamically adaptive quadtree grids. International Journal for Numerical Methods in Fluids, Vol. 46, pp. 127-162. https://doi.org/10.1002/fld.748
  13. Liang, Q., Zang, J., Borthwick, A.G.L., and Taylor, P.H. (2007) Shallow flow simulation on dynamically adaptive cut cell quadtree grids. International Journal for Numerical Methods in Fluids, Vol. 53, pp. 1777-1799. https://doi.org/10.1002/fld.1363
  14. Park, J., Yoon, S., and Kim, C. (2010) Multi-dimensional limiting process for hyperbolic conservation laws on unstructured grids. Journal of Computational Physics, Vol. 229, pp. 788-812. https://doi.org/10.1016/j.jcp.2009.10.011
  15. Popinet, S. (2003) Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries. Journal of Computational Physics, Vol. 190, No. 2, pp. 572-600. https://doi.org/10.1016/S0021-9991(03)00298-5
  16. Popinet, S. (2011) Quadtree-adaptive tsunami modelling. Ocean Dynamics, DOI: 10.1007/s10236-011-0438-z.
  17. Sweby, P.K. (1984) High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM Journal on Numerical Analysis, Vol. 21(5), pp. 995-1011. https://doi.org/10.1137/0721062
  18. Toro, E.F. (2000) Shock-capturing methods for free-surface shallow flow. John Wiley&Sons, LTD, New York.
  19. Yoon, S., Kim, C., and Kim, K. (2008) Multi-dimensional limiting process for three-dimensional flow physics analyses. Journal of Computational Physics, Vol. 227, pp. 6001-6043. https://doi.org/10.1016/j.jcp.2008.02.012