Journal of Soil and Groundwater Environment (한국지하수토양환경학회지:지하수토양환경)

Korean Society of Soil and Groundwater Environment (한국지하수토양환경학회)
권호 표지

Comparison of ELLAM and LEZOOMPC for Developing an Efficient Modeling Technique

효율적인 수치 모델링 기법 개발을 위한 ELLAM과 LEZOOMPC의 비교분석

  • Suk Hee-Jun (Korea Water Resources Corporation, Korea Institute of Water and Environment)
  • 석희준 (한국수자원공사 수자원연구원)
  • Published : 2006.02.01
Download PDF
⟨ Previous Next ⟩

Abstract

This study summarizes advantages and disadvantages of numerical methods and compares ELLAM and LEZOOMPC to develop an efficient numerical modeling technique on contaminant transport. Eulerian-Lagrangian method and Eulerian method are commonly used numerical techniques. However Eulerian-Lagrangian method does not conserve mass globally and fails to treat boundary in a straightforward manner. Also, Eulerian method has restrictions on the size of Courant number and mesh Peclet number because of time truncation error. ELLAM (Eulerian Lagrangian Localized Adjoint Method) which has been popularly used for past 10 years in numerical modeling, is known for overcoming these numerical problems of Eulerian-Lagrangian method and Eulerian method. However, this study investigates advantages and disadvantages of ELLAM and suggests a change for the better. To figure out the disadvantages of ELLAM, the results of ELLAM, LEZOOMPC (Lagrangian-Eulerian ZOOMing Peak and valley Capturing), and visual MODFLOW are compared for four examples having different mesh Peclet numbers. The result of ELLAM generates numerical oscillation at infinite of mesh Peclet number, but that of LEZOOMPC yields accurate simulations. The simulation results suggest that the numerical error of ELLAM could be alleviated by adopting some schemes in LEZOOMPC. In other words, the numerical model which combines ELLAM with backward particle tracking, forward particle tracking, adaptively local zooming, and peak/valley capturing of LEZOOMPC can be developed for not only overcoming the numerical error of ELLAM, but also keeping the numerical advantage of ELLAM.

본 연구는 오염물 거동에 대한 수치해석을 위해 보편적으로 사용되고 있는 수치 방법들의 장단점을 총괄적으로 나타내고, 효율적인 수치모델링 기법 개발을 위해 ELLAM과 LEZOOMPC를 비교분석하였다. 지하수 분야에서 가장 많이 사용되는 수치 방법은 Eulerian-Lagrangian 방식과 Eulerian 방식인데, Eulerian-Lagrangian 방식은 수치영역 내에서 일반적으로 질량을 보존하지 못하고, 경계조건을 체계적으로 처리하지 못하는 한계를 갖고 있다. 반면에 Eulerian 빙식은 시간 및 공간 절삭 오차로 인해서 시간 간격 및 격자 크기를 극히 줄여야 하는 제약을 갖고 있다. 최근 10 년간 지하수 분야에서 크게 대두되고 있는 수치기법인 ELLAM(Eulerian Lagrangian Localized Adjoint Method)은 Eulerian-Lagrangian 방식과 Eulerian 방식에서 나타나는 수치 제약점이나 한계점을 동시에 해결하는 수치기법으로 알려져 왔다. 그러나 본 연구에서는 ELLAM의 장단점을 파악하고 보완점을 제안한다. ELLAM의 단점을 파악하기 위해, mesh Peclet number가 다른 예제들을 설정하고, 그 예제들에 대한 ELLAM, LEZOOMPC(Lagrangian-Eulerian ZOOMing Peak and valley Capturing)와 visual MODFLOW의 수치결과들을 해석해와 비교하였다. Mesh Peclet number가 무한대일 때 ELLAM의 수치결과는 수치진동으로 인해 해석해와 일치하지 않았으나, LEZOOMPC의 수치 결과는 해석해와 일치했다. 위의 결과는 ELLAM의 수치오차가 LEZOOMPC의 특성을 이용하여 개선 및 보완될 수 있는 가능성을 시사해 준다. 따라서 ELLAM에 LEZOOMPC의 후향 입지추적, 전향 입지추적, 선택적 국부 격자 세립화 과정과 최고/최저 농도점 이동 추적 과정을 결합하면 ELLAM의 수치적 장점을 유지하면서 mesh Peclet number에 제약을 받지 않는 효율적인 수치모델링 기법을 개발할 수 있을 것으로 판단된다.

Keywords

References

  1. Baptista, A.M., 1987, Solution of advection-dominated transport by Eulerian-Lagrangian methods using the backward methods of characteristics, Ph.D. thesis, Dep. of Civ. Eng., Mass. Inst. of Technol., Cambridge
  2. Baptista, A.M., Adams, E., and Stolzenbach, K., 1984, Eulerian-Lagrangian analysis of pollutant transport in shallow water, Rep. 296, R.M. Parsons Lab. for Water Resour. and Hydrodyn., Mass. Inst. of Technol., Cambridge
  3. Binning, P.J. and Celia, M.A., 1996, A finite volume Eulerian-Lagrangian localized adjoint method for solution of contaminant transport equations in two-dimensional multi phase flow systems, Water Resour. Res., 32, 103-114 https://doi.org/10.1029/95WR02763
  4. Celia, M.A., 1994, Eulerian-Lagrangian localized adjoint methods for contaminant transport simulations, In Computational Methods in Water Resources X, ed. Alexander Peters et al. Kluwer Academic Press, London, 207-216
  5. Celia, M.A., Russell, T.F., Herrera, I., and Ewing, R.E., 1990, An Eulerian-Lagrangian localized adjoint method for the advection-diffusion equation, Adv. Water Resour., 13, 187-206 https://doi.org/10.1016/0309-1708(90)90041-2
  6. Cheng, J.R., Cheng, H.P., and Yeh, G.T., 1996, A Lagrangian-Eulerian method with adaptively local zooming and peak/valley capturing approach to solve two-dimensional advection-diffusion transport equations, International J. Numerical Methods in Engineering, 39, 987-1016 https://doi.org/10.1002/(SICI)1097-0207(19960330)39:6<987::AID-NME891>3.0.CO;2-V
  7. Douglas, J. and Russell, T.F., 1982, Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures, SIAM J. Numer. Anal., 19,871-885 https://doi.org/10.1137/0719063
  8. Harbaugh, A.W., Banta, E.R., Hill, M.C., and McDonald, M.G., 2000, MODFLOW-2000, The U.S. Geological Survey modular ground water model-User guide to modularization concepts and the ground water flow process, Open-File Report 00-92, US Geological Survey
  9. Healy, R.W. and Russell, T.F., 1993, A finite-volume Eulerian-Lagrangian localized adjoint method for solution of the advection-dispersion equation, Water Resow: Res., 29, 2399-2413 https://doi.org/10.1029/93WR00403
  10. Healy, R. W. and Russell, T.F., 1998, Solution of the advectiondispersion equation in two dimensions by a finite-volume Eulerian-Lagrangian localized adjoint method, Adv. Water Resour., 21(1), 11-26 https://doi.org/10.1016/S0309-1708(96)00033-4
  11. Herrera, I., Ewing, R.E., Celia, M.A., and Russell, T.F., 1993, Eulerian-Lagrangian localized adjoint methods: the theoretical framework, Numer. Meth. PDEs, 9, 431-458 https://doi.org/10.1002/num.1690090407
  12. Konikow, L.F. and Bredehoeft, J.D., 1978, Computer model of two-dimensional solute transport and dispersion in groundwater, Techniques of Water-Resources Investigation of the United Sates Geological Survey, chapter C2, book 7, USGS, Reston, Va
  13. Leonard, B.P., 1988, Universal limiter for transient interpolation modeling of advective transport equations: The ULTIMATE conservative difference scheme, NASA Tech. Memo. 100916
  14. Leonard, B.P. and Mokhtari, S., 1990, Beyond first-order unwinding: The ULTRA-SHARP alternative for non-oscillatory steady-state simulation of convection, lnt. J. Numer. Methods Eng., 30, 729-866 https://doi.org/10.1002/nme.1620300412
  15. Russell, T.F., 1990, Eulerian-Lagrangian localized adjoint methods for advection-dominated problems. In Numerical Analysis, 1989, Pitman Res. Notes Math, Series, Vol. 228, ed. D.F. Griffiths & G.A. Watson. Longman Scientific and Technical, Harlow, U.K., 206-228
  16. Williamson, D.L., and Rasch, P.J., 1988, Two-dimensional semi-Lagrangian transport with shape preserving interpolation, Mon. Weather Rev., 117, 102-109 https://doi.org/10.1175/1520-0493(1989)117<0102:TDSLTW>2.0.CO;2
  17. Yeh, G.T., 1990, A Lagrangian-Eulerian method with zoomable hidden fine-mesh approach to solving advection-dispersion equations, Water Resour. Res., 26(6), 1133-1144 https://doi.org/10.1029/WR026i006p01133
  18. Yeh, G.T., Chang, J.R., and Short, T.E., 1992, An exact peak capturing and oscillation-free scheme to solve advection-dispersion transport equations, Water Resour. Res., 28(11), 2937-2951 https://doi.org/10.1029/92WR01751