Journal of the Korean Society for Industrial and Applied Mathematics

  • 제16권3호
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  • Pages.193-204
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  • 2012
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  • 1226-9433(pISSN)
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  • 1229-0645(eISSN)
한국산업응용수학회 (Korean Society for Industrial and Applied Mathematics)
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NUMERICAL COMPARISON OF WENO TYPE SCHEMES TO THE SIMULATIONS OF THIN FILMS

  • Kang, Myungjoo (DEPARTMENT OF MATHEMATICS, SEOUL NATIONAL UNIVERSITY) ;
  • Kim, Chang Ho (DEPARTMENT OF COMPUTER ENGINEERING, GLOCAL CAMPUS, KONKUK UNIVERSITY) ;
  • Ha, Youngsoo (NATIONAL INSTITUTE FOR MATHEMATICAL SCIENCES)
  • 투고 : 2012.04.20
  • 심사 : 2012.08.13
  • 발행 : 2012.09.25
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초록

This paper is comparing numerical schemes for a differential equation with convection and fourth-order diffusion. Our model equation is $h_t+(h^2-h^3)_x=-(h^3h_{xxx})_x$, which arises in the context of thin film flow driven the competing effects of an induced surface tension gradient and gravity. These films arise in thin coating flows and are of great technical and scientific interest. Here we focus on the several numerical methods to apply the model equation and the comparison and analysis of the numerical results. The convection terms are treated with well known WENO methods and the diffusion term is treated implicitly. The diffusion and convection schemes are combined using a fractional step-splitting method.

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참고문헌

  1. A.L. Bertozzi, A. Munch, M. Shearer, Undercompressive shocks in thin film flows, Physica D 134 431-464 (1999). https://doi.org/10.1016/S0167-2789(99)00134-7
  2. R. Borges, M. Carmona, B. Costa, and W.S. Don, An improved WENO scheme for hyperbolic conservation laws, J. Comput. Phys. 227, 3191-3211 (2008). https://doi.org/10.1016/j.jcp.2007.11.038
  3. Y. HA, Y.-J. KIM, AND T.G. MYERS, On the numerical solution of a driven thin film equation, J. Comput. Phys. 227, 7246-7263 (2008). https://doi.org/10.1016/j.jcp.2008.04.007
  4. Y. Ha, C.L. Gardner, A. Gelb, and C.W. Shu Numerical Simulation of High Mach Number Astrophysical Jets with Radiative Cooling JSC 24, 29-44 (2005).
  5. A. Harten, P.D. Lax, B. van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev. 25, 35 (1983). https://doi.org/10.1137/1025002
  6. A. Harten, On a Class of High Resolution Total-Variation-Stable Finite-Difference Schemes, SIAM J. Numer. Anal.,Vol. 21, no. 1, 1-23 (1984). https://doi.org/10.1137/0721001
  7. A. Harten and G. Zwas, Self-Adjusting Hybrid Schemes for Shock Computations, J. Comput. Phys. 9, 568-583 (1973). https://doi.org/10.1016/0021-9991(72)90012-5
  8. A.K. Henrick, T.D. Aslam, and J.M. Powers, Mapped weighted-essentially-non-oscillatory schemes : achieving optimal order near critical points, J. Comput. Phys. 207, 542-567 (2005). https://doi.org/10.1016/j.jcp.2005.01.023
  9. G-S. Jiang and C.-W. Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126 (1996), pp. 202-228. https://doi.org/10.1006/jcph.1996.0130
  10. P. D. Lax and B.Wendroff, Systems of conservation laws, Communications in Pure and Applied Mathematics 13, 217(1960). https://doi.org/10.1002/cpa.3160130205
  11. X-D. Liu, S. Osher, and T. Chan,Weighted essentially non-oscillatory schemes, J. Comput. Phys., 115 (1994), pp. 200-212. https://doi.org/10.1006/jcph.1994.1187
  12. R. J. LeVeque, Numerical Methods for Conservation Laws, Birkhauser Verlag, Basel (1992).
  13. P. L. Roe, Approximate Riemann solvers, paremeter vectors, and difference schemes, J.Comp. Phys.,43, 357-372(1981). https://doi.org/10.1016/0021-9991(81)90128-5
  14. T. Schmutzler and W. M. Tscharnuter, Effective radiative cooling in optically thin plasmas, Astronomy and Astrophysics 273, 318-330, (1993).
  15. C.-W. Shu, Total-variation-diminshing time discretizations SIAM J. Sci. Statist. Comput. 9, 1073-1084 (1988). https://doi.org/10.1137/0909073
  16. C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock capturing schemes, J. Comput. Phys. 77, 32 (1988).
  17. C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock capturing schemes,II, J. Comput. Phys. 83, 32-78 (1989). https://doi.org/10.1016/0021-9991(89)90222-2
  18. C.W. Shu, ENO and WENO schemes for hyperbolic conservation laws, in: B. Cockburn, C. Johnson, C.W. Shu, E. Tadmor (Eds.), Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics, vol. 1697, Springer, Berlin, 1998, pp. 325-432 (also NASA CR- 97-206253 and ICASE-97-65 Rep., NASA Langley Research Center, Hampton [VA, USA]).
  19. P. K. Sweby, High resolution schemes using flux limiters hyperbolic conservation laws, SIAM J.Numer. Anal. Vol. 21, No.5, 995-1011(1984). https://doi.org/10.1137/0721062
  20. E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer-Verlag , New York (1997).
  21. R.J. LEVEQUE, Numerical methods for conservation laws, Lectures in Mathematics ETH Zurich, Birkhauser Verlag, Basel, 1990.
  22. R.J. LEVEQUE, High-resolution conservative algorithms for advection in incompressible flow. SIAM J. Sci. Compu., 1995.
  23. R.J. LEVEQUE, Clawpack Version 4.0 Users Guide, Technical report, University of Washington, Seattle, 1999. Available online at http://www.amath.washington.edu/claw/.
  24. H. NESSYAHU AND E. TADMOR, Nonoscillatory central differencing for hyperbolic conservation laws, J. Comput. Phys., 87 (1990), no. 2, 408-463. https://doi.org/10.1016/0021-9991(90)90260-8
  25. B. VAN LEER, MUSCL, A New Approach to Numerical Gas Dynamics. In Computing in Plasma Physics and Astrophysics, Max-Planck-Institut fur Plasma Physik, Garchung, Germany, April 1976.
  26. A.L. BERTOZZI, A. MUNCH, M. SHEARER, Undercompressive shocks in thin film flows Physica D 134 (1999) 431-464 https://doi.org/10.1016/S0167-2789(99)00134-7