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The equilibrium state method for hyperbolic conservation laws with stiff reaction terms. (English) Zbl 1349.80042

Summary: A new fractional-step method is proposed for numerical simulations of hyperbolic conservation laws with stiff source terms arising from chemically reactive flows. In stiff reaction problems, a well-known spurious numerical phenomenon, the incorrect propagation speed of discontinuities, may occur in general fractional-step algorithm due to the underresolved numerical solution in both space and time. The basic idea of the present proposed scheme is to replace the cell average representation with a two-equilibrium states reconstruction during the reaction step, which allows us to obtain the correct propagation of discontinuities for stiff reaction problems in an underresolved mesh. Because the definition of these two-equilibrium states for each transition cell is independent of its neighboring cells, the proposed method can be extended to multi-dimensional problems directly. In addition, this method is promising to deal with more complicated real-world problems after being extended to multi-species/multi-reactions system. Extensive numerical examples for one- and two-dimensional scalar and Euler system demonstrate the reliability and robustness of this novel method.

MSC:

80M20 Finite difference methods applied to problems in thermodynamics and heat transfer
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
80A32 Chemically reacting flows
92E20 Classical flows, reactions, etc. in chemistry

Software:

AUSM
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References:

[1] Colella, P.; Majda, A.; Roytburd, V., Theoretical and numerical structure for numerical reacting waves, SIAM J. Sci. Stat. Comput., 7, 1059-1079 (1986) · Zbl 0633.76060
[2] LeVeque, R. J.; Yee, H. C., A study of numerical methods for hyperbolic conservation laws with stiff source terms, J. Comput. Phys., 86, 187-210 (1990) · Zbl 0682.76053
[3] Nguyen, D.; Gibou, F.; Fedkiw, R., A fully conservative ghost fluid method & stiff detonation waves, (Proceedings of the 12th International Detonation Symposium. Proceedings of the 12th International Detonation Symposium, S. Diego, CA (2002))
[4] Bourlioux, A.; Majda, A.; Roytburd, V., Theoretical and numerical structure for unstable one-dimensional detonations, SIAM J. Appl. Math., 51, 303-343 (1991) · Zbl 0731.76076
[5] Jeltsch, R.; Klingenstein, P., Error estimators for the position of discontinuities in hyperbolic conservation laws with source term which are solved using operator splitting, Comput. Vis. Sci., 1, 231-249 (1999) · Zbl 0970.65100
[6] Bihari, B.; Schwendeman, D., Multiresolution schemes for the reactive Euler equations, J. Comput. Phys., 154, 197-230 (1999) · Zbl 0959.76049
[7] Chorin, Random choice solution of hyperbolic systems, J. Comput. Phys., 22, 517-533 (1976) · Zbl 0354.65047
[8] Chorin, Random choice methods with applications for reacting gas flows, J. Comput. Phys., 25, 253-272 (1977) · Zbl 0403.65049
[9] Majda, A., Numerical study of the mechanisms for initiation of reacting shock waves, SIAM J. Sci. Stat. Comput., 11, 950-974 (1990) · Zbl 0699.76075
[10] Engquist, B.; Sjögreen, B., Robust difference approximations of stiff inviscid detonation waves (1991), UCLA, Technical Report CAM 91-03
[11] Berkenbosch, A.; Kaasschieter, E.; Klein, R., Detonation capturing for stiff combustion chemistry, Combust. Theory Model., 2, 313-348 (1998) · Zbl 0936.80005
[12] Helzel, C.; LeVeque, R.; Warneke, G., A modified fractional step method for the accurate approximation of detonation waves, SIAM J. Sci. Stat. Comput., 22, 1489-1510 (1999) · Zbl 0983.65105
[13] Bao, W.; Jin, S., The random projection method for hyperbolic conservation laws with stiff reaction terms, J. Comput. Phys., 163, 216-248 (2000) · Zbl 0966.65073
[14] Bao, W.; Jin, S., The random projection method for stiff detonation capturing, SIAM J. Sci. Comput., 23, 1000-1025 (2001) · Zbl 0997.80013
[15] Bao, W.; Jin, S., The random projection method for stiff multispecies detonation capturing, J. Comput. Phys., 178, 37-57 (2002) · Zbl 1017.76073
[16] Tosatto, L.; Vigevano, L., Numerical solution of under-resolved detonations, J. Comput. Phys., 227, 2317-2343 (2008) · Zbl 1141.80013
[17] Wang, W.; Shu, C. W., High order finite difference methods with subcell resolution for advection equations with stiff source terms, J. Comput. Phys., 231, 190-214 (2012) · Zbl 1457.65064
[18] Yee, H. C.; Kotov, D. V.; Wang, Wei; Shu, Chi-Wang, Spurious behavior of shock-capturing methods by the fractional step approach: Problems containing stiff source terms and discontinuities, J. Comput. Phys., 241, 266-291 (2013) · Zbl 1349.80048
[19] Strang, G., On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5, 506-517 (1968) · Zbl 0184.38503
[20] Kotov, D.; Yee, H.; Wang, W.; Shu, C. W., On spurious numerics in solving reactive equations, (Proceedings of the ASTRONUM-2012. Proceedings of the ASTRONUM-2012, The Big Island, Hawaii (2012))
[21] Dumbser, M.; Enaux, C.; Toro, E. F., Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws, J. Comput. Phys., 8, 3971-4001 (2008) · Zbl 1142.65070
[22] Hidalgo, A.; Dumbser, M., ADER schemes for nonlinear systems of stiff advection-diffusion-reaction equations, J. Sci. Comput., 48, 173-189 (2011) · Zbl 1221.65231
[23] Li, J. G., Propagation of ocean surface waves on a spherical multiple-cell grid, J. Comput. Phys., 231, 8262-8277 (2012)
[24] Li, J. G., Upstream non-oscillatory advection schemes, Mon. Weather Rev., 136, 4709-4729 (2008)
[25] Zalesak, S. T., Fully multi-dimensional flux-corrected transport algorithms for fluids, J. Comput. Phys., 31, 335-362 (1979) · Zbl 0416.76002
[26] Liou, M. S.; Christopher, J., A new flux splitting schemes, J. Comput. Phys., 107, 23-39 (1993) · Zbl 0779.76056
[27] LeVeque, R., Numerical Method for Conservation Laws (1992), Birkhäuser
[28] Liou, M. S., A Sequel to AUSM: \(AUSM^+\), J. Comput. Phys., 129, 364-382 (1996) · Zbl 0870.76049
[29] Kailasanath, K.; Oran, E. S., Determination of detonation cell size and the role of transverse waves in two-dimensional detonation, Combust. Flame, 61, 199-209 (1985)
[30] Wang, C.; Zhang, D. L.; Jiang, Z. L., Numerical investigation of detonation sweeping an interface of inert gas and its decoupling, Explos. Shock Waves, 26, 6, 556-561 (2006)
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