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High Order Finite Difference Methods with Subcell Resolution for Stiff Multispecies Discontinuity Capturing

Published online by Cambridge University Press:  22 January 2015

Wei Wang ,
Chi-Wang Shu ,
H.C. Yee ,
Dmitry V. Kotov  and
Björn Sjögreen
Wei Wang*
Affiliation:
Department of Mathematics and Statistics, Florida International University, Miami, FL 33199, USA
Chi-Wang Shu
Affiliation:
Division of Applied Mathematics, Brown University, Providence, RI 02912, USA
H.C. Yee
Affiliation:
NASA Ames Research Center, Moffett Field, CA 94035, USA
Dmitry V. Kotov
Affiliation:
Bay Area Environmental Research Institute, 625 2nd St. Ste 209 Petaluma, CA 94952, USA
Björn Sjögreen
Affiliation:
Lawrence Livermore National Laboratory, Livermore, CA 94551, USA
*
*Email addresses: weiwang1@fiu.edu (W. Wang), shu@dam.brown.edu (C.-W. Shu), helen.m.yee@nasa.gov (H. C. Yee), dmitry.v.kotov@nasa.gov (D. V. Kotov), sjogreen2@llnl.gov (B. Sjögreen)
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Abstract

In this paper, we extend the high order finite-difference method with subcell resolution (SR) in [34] for two-species stiff one-reaction models to multispecies and multireaction inviscid chemical reactive flows, which are significantly more difficult because of the multiple scales generated by different reactions. For reaction problems, when the reaction time scale is very small, the reaction zone scale is also small and the governing equations become very stiff. Wrong propagation speed of discontinuity may occur due to the underresolved numerical solution in both space and time. The present SR method for reactive Euler system is a fractional step method. In the convection step, any high order shock-capturing method can be used. In the reaction step, an ODE solver is applied but with certain computed flow variables in the shock region modified by the Harten subcell resolution idea. Several numerical examples of multispecies and multireaction reactive flows are performed in both one and two dimensions. Studies demonstrate that the SR method can capture the correct propagation speed of discontinuities in very coarse meshes.

Keywords

65M0676V0580A32Stiff reaction termshock capturingdetonationWENOENO subcell resolutionmultispeciesmultireactions
Type
Research Article
Information
Communications in Computational Physics , Volume 17 , Issue 2 , February 2015 , pp. 317 - 336
Copyright
Copyright © Global Science Press Limited 2015 

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References

[1]Bao, W. and Jin, S.The random projection method for hyperbolic conservation laws with stiff reaction terms. J. Comput. Phys., 163:216–248, 2000.Google Scholar
[2]Bao, W. and Jin, S.The random projection method for stiff detonation capturing. SIAM J. Sci. Comput., 23:1000–1025, 2001.CrossRefGoogle Scholar
[3]Bao, W. and Jin, S.The random projection method for stiff multispecies detonation capturing. J. Comput. Phys., 178:37–57, 2002.Google Scholar
[4]Ben-Artzi, M.The generalized Riemann problem for reactive flows. J. Comput. Phys., 81:70–101, 1989.Google Scholar
[5]Berkenbosch, A.Kaasschieter, E. and Klein, R.Detonation capturing for stiff combustion chemistry. Combust. Theory Model., 2:313–348, 1998.CrossRefGoogle Scholar
[6]Bihari, B. and Schwendeman, D.Multiresolution schemes for the reactive euler equations. J. Comput. Phys., 154:197–230, 1999.CrossRefGoogle Scholar
[7]Bourlioux, A.Majda, A. and Roytburd, V.Theoretical and numerical structure for unstable one-dimensional detonations. SIAM J. Appl. Math., 51:303–343, 1991.Google Scholar
[8]Chang, S.-H.On the application of subcell resolution to conservation laws with stiff source terms. NASA Technical Memorandum 102384, ICOMP Report 89–27, 1989.Google Scholar
[9]Chang, S.-H.On the application of subcell resolution to conservation laws with stiff source terms. NASA Lewis Research Center, Computational Fluid Dynamics Symposium on Aero-propulsion, 215–225, 1991.Google Scholar
[10]Chorin, A.Random choice solution of hyperbolic systems. J. Comput. Phys., 22:517–533, 1976.Google Scholar
[11]Chorin, A.Random choice methods with applications for reacting gas flows. J. Comput. Phys., 25:253–272, 1977.Google Scholar
[12]Colella, P.Majda, A. and Roytburd, V.Theoretical and numerical structure for numerical reacting waves. SIAM J. Sci. Stat. Comput., 7:1059–1080, 1986.Google Scholar
[13]Engquist, B. and Sjögreen, B.Robust difference approximations of stiff inviscid detonation waves. Technical Report CAM 91–03, UCLA., 1991.Google Scholar
[14]Griffiths, D.Stuart, A. and Yee, H.C.Numerical wave propagation in an advection equation with a nonlinear source term. SIAM J. Numer. Anal., 29:1244–1260, 1992.Google Scholar
[15]Harten, A.ENO schemes with subcell resolution. J. Comput. Phys., 83:148–184, 1989.Google Scholar
[16]Helzel, C.LeVeque, R. and Warneke, G.A modified fractional step method for the accurate approximation of detonation waves. SIAM J. Sci. Stat. Comput., 22:1489–1510, 1999.CrossRefGoogle Scholar
[17]Hidalgo, A. and Dumbser, M.ADER schemes for nonlinear systems of stiff advection-diffusion-reaction equations. J. Sci. Comput., 48:173–189, 2011.Google Scholar
[18]Hu, X.Y.Adams, N.A. and Shu, C.-W.Positivity-preserving flux limiters for high-order conservative schemes solving compressible Euler equations. J. Comput. Phys., 242:169–180, 2013.Google Scholar
[19]Jeltsch, R. and 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.Google Scholar
[20]Jiang, G and Shu, C.-W.Efficient implementation of weighted ENO schemes. J. Comput. Phys., 126:202–228, 1996.Google Scholar
[21]Kotov, D.Yee, H.C.Panesi, M.Wray, A.Prabhu, D.1D and 2D Simulation Related to the NASA Electric Arc Shock Tube Experiments, 21st AIAA Computational Fluid Dynamics Conference, contributed paper, June 24–27, 2013, San Diego, CA.Google Scholar
[22]LeVeque, R.J. and Yee, H.C.A study of numerical methods for hyperbolic conservation laws with stiff source terms. J. Comput. Phys., 86:187–210, 1990.Google Scholar
[23]LeVeque, R.J. and Shyue, K.-M.One-dimensional front tracking based on high resolution wave propagation methods. SIAM J. Sci. Comput., 16:348–377, 1995.CrossRefGoogle Scholar
[24]Majda, A. and Roytburd, V.Numerical study of the mechanisms for initiation of reacting shock waves. SIAM J. Sci. Stat. Comput., 11:950–974, 1990.Google Scholar
[25]Miniati, F. and Colella, P.A modified higher order Godunov scheme for stiff source conservative hydrodynamics. J. Comput. Phys., 224:519–538, 2007.CrossRefGoogle Scholar
[26]Nguyen, D.Gibou, F. and Fedkiw, R.A fully conservative ghost fluid method & stiff detonation waves. Proceedings of the 12th International Detonation Symposium, San Diego, CA, 2002.Google Scholar
[27]Pember, R.Numerical methods for hyperbolic conservation laws with stiff relaxation, I. spurious solutions. SIAM J. Appl. Math., 53:1293–1330, 1993.Google Scholar
[28]Shu, C.-W. and Osher, S.Efficient implementation of essentially non-oscillatory shock capturing schemes, II. J. Comput. Phys., 83:32–78, 1989.CrossRefGoogle Scholar
[29]Strang, G.On the construction and comparison of difference schemes. SIAM J. Numer. Anal., 5:506–517, 1968.Google Scholar
[30]Sun, Y. and Engquist, B.Heterogeneous multiscale methods for interface tracking of combustion fronts. Multiscale Model. Simul., 5:532–563, 2006.Google Scholar
[31]Ton, V.Improved shock-capturing methods for multicomponent and reacting flows. J. Comput. Phys., 128:237–253, 1996.Google Scholar
[32]Tosatto, L. and Vigevano, L.Numerical solution of under-resolved detonations. J. Comput. Phys., 227:2317–2343, 2008.CrossRefGoogle Scholar
[33]Wang, W.Shu, C.-W.Yee, H.C. and Sjögreen, B.High order finite difference methods with subcell resolution for advection equations with stiff source terms. J. Comput. Phys., 231:190–214, 2012.Google Scholar
[34]Yee, H.C.Kotov, D.V.Wang, W. and Shu, C.-W.Spurious behavior of shock-capturing methods: problems containing stiff source terms and discontinuities. J. Comput. Phys., 241:266–291, 2013.CrossRefGoogle Scholar