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A Local Extrapolation Method for Hyperbolic Conservation Laws. I. The ENO Underlying Schemes

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Abstract

In this paper, we introduce a local extrapolation method (LEM) for the essentially non-oscillatory (ENO) schemes solving nonlinear hyperbolic conservation laws. The method extrapolates the numerical flux of the underlying scheme so that it keeps conservativity. We use a minmod type limiter to avoid spurious oscillations. We propose a new balancing technique that preserves the symmetry of a symmetric wave that works well for a wide range of CFL numbers. We also introduce two artificial compression procedures to the LEM which yield sharp resolutions of contact discontinuities. Numerical examples are presented to illustrate the performance of the method.

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REFERENCES

  1. Boris, J. P., and Books, D. L. (1973). Flux-corrected transport I:SHAS TA, a fluid trans-port algorithm that works. J. Compute. Pays. 11, 38–69.

    Google Scholar 

  2. Cern, I.-L., Glimm, J., McBryan, U., Plohr, B., and Yaniv, S. (1986). Front tracking for gas dynamics. J. Comput. Phys. 62, 83–110.

    Google Scholar 

  3. Chern, I.-L., and Collela, P. (1986). A conservative front tracking method for hyperbolic conservation laws, preprint.

  4. Cockburn, B., Lin, S.-Y., and Shu C.-W. (1989). TVB Runge-Kutta local projection dis-continuous Galerkin finite element method for conservation laws. III. One-dimensional systems. J. Comput. Phys. 84, No. 1, 90–113.

    Google Scholar 

  5. Cockburn, B., Hou S., and Shu, C.-W. (1990). The Runge_Kutta local projection discon-tinuous Galerkin finite element method for conservation laws. IV. The multidimensional case. Math. Comput. v54, 545–581.

    Google Scholar 

  6. Glimm, J., Grove, J., Lindquist, B., McBryan, O., and Tryggvanson, G. (1988). The bifur-cation of tracked scalar wave. SIAM J. Sci. Statist. Comput. 7, 61–79.

    Google Scholar 

  7. Goodman J. B., and LeVeque, R. J. (1988). A geometric approach to high resolution TVD schemes. SIAM J. Numer. Anal. 25, 268–284.

    Google Scholar 

  8. Harten, A. (1991). Recent developments in shock-capturing schemes. In Proceedings of the International Congress of Mathematicians, Vols. I and II (Kyoto, 1990), pp. 1549–1559, Math. Soc. Japan, Tokyo.

    Google Scholar 

  9. Harten, A. (1983). High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49, 357–393.

    Google Scholar 

  10. Harten, A. (1978). The artificial compression method for computation of shocks and con-tact discontinuities: III. Self-adjusting hybrid schemes. Math. Comp. 32, 363–389.

    Google Scholar 

  11. Harten, A. (1989). ENO schemes with subcell resolution. J. Comp. Phys. 83, 148–184.

    Google Scholar 

  12. Harten, A. (1995). Multiresolution algorithms for the numerical solution of hyperbolic conservation laws. Comm. Pure Appl. Math. 48, No. 12, 1305–1342.

    Google Scholar 

  13. Harten, A., and Osher, S. (1987). Uniformly high order accurate non-oscillatory schemes, I. SIAM J. Numer. Anal. 24, 279.

    Google Scholar 

  14. Harten, A., Osher, S., Engquist, B., and Chakravarthy, S. (1986). Some results on uniformly high order accurate essentially non-oscillatory schemes. J. Appl. Numer. Math. 2, 347.

    Google Scholar 

  15. Harten, A., Engquist, B., Osher, S., and Chakravarthy, S. (1987). Uniformly high order accurate essentially non-oscillatory schemes, III. J. Comput. Phys. 71, 231.

    Google Scholar 

  16. Hannappel, R., Hauser, T., and Friedrich, R. (1995). A comparison of ENO and TVD schemes for the computation of shock-turbulence interaction. J. Comput. Phys. 121, 176–184.

    Google Scholar 

  17. Huynh, H. T. (1995). Accurate upwind methods for the Euler equations. SIAM J. Numer. Anal. 32, 1565–1620.

    Google Scholar 

  18. Iserles, A., and Strang, G. (1983). The optimal accuracy of difference schemes. Trans. Amer. Math. Soc. 277, 779.

    Google Scholar 

  19. Jeng, J.-H., and Payne, U.-J. (1995). An adaptive TVD limiter. J. Comput. Phys. 118, 229–241.

    Google Scholar 

  20. Jiang, G.-S., and Shu, C.-W. (1996). Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, No. 1, 202–228.

    Google Scholar 

  21. Jin, B. X. (1993). An artificial compression method for the computation of contact discon-tinuity. Comp. Math. (China.) 1, 121.

    Google Scholar 

  22. Lappaz, T., Leonard, A., and Dimotakis, P. E. (1993). An adaptive Lagrangian method for computing 1D reacting and non-reacting flows. J. Comput. Phys. 104, 361–376.

    Google Scholar 

  23. Lavery, J. E. (1993). Capturing contact discontinuities in steady-state conservation laws. J. Comput. Phys. 108, 59–72.

    Google Scholar 

  24. Lax, P. D., and Wendroff, B. (1990). Systems of Conservation Laws, Comm. Pure Appl. Math., Vol. 13 (1990), pp. 217–237.

    Google Scholar 

  25. LeVeque, J., and Shyue, K. M. (1992). Shock tracking based on high resolution wave propagation methods, Technical Report 29–3, University of Washington, Seattle.

    Google Scholar 

  26. Li, X. L., Jin, B. X., and Glimm, J. (1995). Numerical study for the three dimensional Rayleigh-Taylor instability through the TVD/AC scheme and parallel computation. SUNYSB-AMS-95–03.

  27. Liem, C. B., Lü, T., and Shih, T. M. (1995). The splitting extrapolation method. A new technique in numerical solution of multidimensional problems. With a preface by Zhong-ci Shi, Series on Applied Mathematics, 7. World Scientific Publishing Co., Inc., River Edge, NJ.

    Google Scholar 

  28. Zhu, Q.-D., and Lin, Q. (1989). The hyperconvergence theory of finite elements, (Chinese) Hunan Science and Technology Publishing House, Changsha.

    Google Scholar 

  29. Liu, X.-D., and Osher, S. (1996). Nonoscillatory high order accurate self-similar maxi-mum principle satisfying shock capturing schemes. I. SIAM J. Numer. Anal. 33, No. 2, 760–779.

    Google Scholar 

  30. Liu, X.-D., Osher, S., and Chan, T. (1994). Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115, 200–212.

    Google Scholar 

  31. Loh, C. Y., and Hui, W. H. (1990). A new Lagrangian method for steady supersonic flow computation. J. Comput. Phys. V89, No. 1.

    Google Scholar 

  32. Mao, D. (1993). A treatment of discontinuities for finite difference methods in the two-dimensional case. J. Comput. Phys. 104, 377–397.

    Google Scholar 

  33. Perthame, B. (1992). Second-order Boltzmann schemes for compressible Euler equations in one and two space dimensions. SIAM J. Numer. Anal. 29, No. 1, 1–19.

    Google Scholar 

  34. Richtmyer, R. D., and Morton, K. W. (1967). Difference methods for initial-value problems, Wiley-Interscience.

  35. Roe, P. L. (1981). Numerical algorithms for the linear wave equations, Royal Aircraft Establishment Technical Report 81047.

  36. Roe, P. L. (1991). Discontinuous solutions to hyperbolic systems under operator splitting. Numer. Met. Part. Diff. Equ. 7, 277–297.

    Google Scholar 

  37. Shu, C.-W. (1987). TVB uniformly high-order schemes for conservation laws. Math. Comp. 49, No. 179, 105–121.

    Google Scholar 

  38. Shu, C.-W., and Osher, S. (1989). Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comp. Phys. 83, 32.

    Google Scholar 

  39. Sweby, P. (1984). High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal. 21, 995.

    Google Scholar 

  40. Van Leer, B. (1974). Towards the ultimate conservative differrence schemes II. Monotonicity and conservation combined in a second order scheme. J. Comput. Phys. 14, 361–370.

    Google Scholar 

  41. Van Leer, B. (1979). Towards the ultimate conservative differrence schemes V. A second order sequel to Godunov's method. J. Comput. Phys. 32, 101–136.

    Google Scholar 

  42. Woodward, P., and Colella, P. (1984). The piecewise-parabolic method (P PM) for gas-dynamical simulations. J. Comput. Phys. 54, 115.

    Google Scholar 

  43. Wu, L. (1997). Linear stability of the local extrapolation method for conservative approximation to the advective equation, Thesis, Kansas State University.

  44. Yang, H. (1990). An artificial compression method for ENO schemes: the slope modifica-tion method. J. Comput. Phys. V 89, No. 1, 125.

    Google Scholar 

  45. Zhu, Y. L., Zhong, X. C., Chen, B. M., and Zhang, Z. M. (1980). Difference methods for initial-boundary problems and flows around bodies, Science Press, Beijing, China.

    Google Scholar 

  46. Zalesak, S. T. (1987). A preliminary comparison of modern shock-capturing schemes: Linear advections. In Vichnevetsky, R., and Stepleman, R. S. (eds.), Advances in Computer Methods for Partial Diferential Equations, VI, IMACS, pp. 15–22.

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  1. Department of Mathematics, Kansas State University, Manhattan, Kansas, 66506

    Huanan Yang

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Yang, H. A Local Extrapolation Method for Hyperbolic Conservation Laws. I. The ENO Underlying Schemes. Journal of Scientific Computing 15, 231–264 (2000). https://doi.org/10.1023/A:1007685827323

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  • DOI: https://doi.org/10.1023/A:1007685827323