Abstract
In this paper, we further analyze, improve and develop the conservative front-tracking method developed in Mao(1995, SIAM. J. Numer. Anal., 32, 1677–1703). The method in Mao(1995, SIAM. J. Numer. Anal., 32, 1677–1703) is only first-order accurate near the tracked discontinuities when applied to systems of conservation laws. This is because in the system case near tracked discontinuities there are waves in other characteristic fields and they may propagate across the discontinuities. The method in Mao(1995, SIAM. J. Numer. Anal., 32, 1677–1703) treats this propagate-across of waves in a first-order fashion. In this paper, we develop a high-order treatment of this wave propagate-across on tracked discontinuities and thus enhance the accuracy of the method. We present a rigorous analysis of truncation error to show that the method equipped with the developed treatment of wave propagate-across is high-order accurate in a certain sense. We also discuss some matters concerning the application of the method to the Euler system of gas dynamics, such as the treatment of reflection wall boundaries, application to the multifluid flows, and the programming of the algorithm. Numerical examples are presented to show the efficiency of the method.
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Aslam T.D. (2003). A level set algorithm for tracking discontinuities in hyperbolic conservation laws II: system of equations. J. Sci. Comput. 19, 37–62
Aslam T.D. (2001). A level set algorithm for tracking discontinuities in hyperbolic conservation laws I: scalar equations. J. Comput. Phys. 167(2): 413–438
Chern, I-L., and Colella, P. (1987). A conservative front tracking method for hyperbolic system of conservation laws, LLNL Rep. No. UCRL-97200 (1987)
Colella P., Woodward P.R. (1984). The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput Phys. 54, 115–173
Glimm J., Li Xiao-lin., Liu Ying-jie., Xu Zhi-liang., Zhao Ning. (2003). Conservative front-tracking with improved accuracy. SIAM J. Numer. Anal. 41(5): 1926–1947
Henshaw W.D. (1987). A scheme for numerical solution of hyperbolic system of conservation laws. J. Comput. Phys 68, 25–47
Harten A. (1989). ENO schemes with subcell resolution. J Comput. Phys. 83, 148–184
Harten A., Engquist B., Osher S., Chakravarthy S.R (1987). Uniformly high-order accurate essentially non-oscillatory scheme, III. J. Comput. Phys. 71, 231–303
Karni S. (1994). Multicomponent flow calculations by a consistent primitive algorightm. J. Comput. Phys. 112, 31–43
Lax P.D. (1957). Hyperbolic system of conservation laws. II J. Comm. Pure Appl. Math. 10, 537–566
LeVeque R.J., Shyue K.M. (1996). Two dimensional front tracking based on high resolution wave progation methods. J.Comput. Phys. 123, 354–368
LeVeque R.J., Shyue K.M. (1995). One dimensional front tracking based on high resolution wave propogation methods. SIAM J. Sci. Comput. 16, 348–377
LeVeque R.J. (1990). Numerical Methods for Conservation Laws. Birkhauser-verlag, Basel, Boston, Berlin
LeVeque, R. J. (2000). Finite Volume Methods for Hyperbolic Problems, Published by the press Syndicate of the University of Cambridge
Mao D. (2000). Towards front tracking based on conservation in two space dimension. SIAM J. SCI. Comput. 22, 113–151
Mao D. (1995). A shock tracking technique based on conservation in one space dimmension. SIAM J. Numer. Anal. 32, 1677–1703
Mao D. (1992). A treatment of discontinuities for finite difference methods. J. Comput. Phys. 103, 359–369
Mao D. (1991). A Treatment of discontinuities in shock-capturing finite difference methods. J. Comput. Phys. 92, 422–455
Mao, D. (1991). A treatment for discontinuities. In Engquist B., and Gustafsson, B., (eds.), (Proc. Third International Conference on Hyperbolic Problem, Uppsala, Swedem, 1990), Studentlitteratur, Sweden
Mao D. (1985). A difference scheme for shock calculation J. Comput. Math. 3, 356–382(in Chinese)
Moretti, (1972). Thoughts and afterthoughts about shock computations, Report No. PIBAL-72-37, PolytechnicInstitute of Brooklyn
Richtmyer R.D., Morton K.W. (1967). Difference Methods for Initial-value Problems. Interscience publishers, New York London Sydney
Roe P.L. Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43, 357–372
Shyue Keh-Ming. (1998). An efficient shock-capturing algorithm for compressible multicomponent problems. J. Comput. Phys. 142, 208–242
Shu Chi-wang., Osher S. (1989). Efficient implementation of essentially non-oscillatory shock-capturing schemes, II. J Comput. Phys. 83, 32–78
Shu Chi-Wang. (1987). TVB boundary treatment for numerical solutions of conservation laws. Math. Comp. 49, 123–134
Sod G.A. (1978). A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J Comput. Phys. 27, 1–31
Swartz B.K., Wendroff B. (1986). A front tracking code based on Godunov’s method. Appl. Numer. Math. 2, 385–397
LeVeque R.J., Shyue K.M. (1995). One dimensional front tracking based on high resolution wave propogation methods. SIAM J. Sci. Comput. 16, 348–377
LeVeque R.J. (1990). Numerical Methods for Conservation Laws. Birkhauser-verlag, Basel, Boston, Berlin
LeVeque R. J. (2000). Finite Volume Methods for Hyperbolic Problems, Published by the press Syndicate of the University of Cambridge
Mao D. (2000). Towards front tracking based on conservation in two space dimension. SIAM J. SCI. Comput. 22, 113–151
Mao D. (1995). A shock tracking technique based on conservation in one space dimmension. SIAM J. Numer. Anal. 32, 1677–1703
Mao D. (1992). A treatment of discontinuities for finite difference methods. J. Comput. Phys. 103, 359–369
Mao D. (1991). A Treatment of discontinuities in shock-capturing finite difference methods. J. Comput. Phys. 92, 422–455
Mao, D. (1991). A treatment for discontinuities. In Engquist B., and Gustafsson, B., (eds.), (Proc. Third International Conference on Hyperbolic Problem, Uppsala, Swedem, 1990), Studentlitteratur, Sweden
Mao D. (1985). A difference scheme for shock calculation J. Comput. Math. 3, 356–382 (in Chinese)
Moretti, (1972). Thoughts and afterthoughts about shock computations, Report No. PIBAL-72-37, PolytechnicInstitute of Brooklyn
Richtmyer R.D., Morton K.W. (1967). Difference Methods for Initial-value Problems. Interscience publishers, New York London Sydney
Roe P.L. Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43, 357–372
Shyue Keh-Ming. (1998). An efficient shock-capturing algorithm for compressible multicomponent problems. J. Comput. Phys. 142, 208–242
Shu Chi-wang., Osher S. (1989). Efficient implementation of essentially non-oscillatory shock-capturing schemes. II. J Comput. Phys. 83, 32–78
Shu Chi-Wang. (1987). TVB boundary treatment for numerical solutions of conservation laws. Math. Comp. 49, 123–134
Sod G.A. (1978). A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J Comput. Phys. 27, 1–31
Swartz B.K., Wendroff B. (1986). A front tracking code based on Godunov’s method. Appl. Numer. Math. 2, 385–397
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Ams(MOS) Subject Classification: 65M05, 65M10, 35L65
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Yan, L., De-kang, M. Further Development of a Conservative Front-Tracking Method for Systems of Conservation Laws in One Space Dimension. J Sci Comput 28, 85–119 (2006). https://doi.org/10.1007/s10915-005-9008-1
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DOI: https://doi.org/10.1007/s10915-005-9008-1