Skip to main content
×
×
Home

An Approach to Obtain the Correct Shock Speed for Euler Equations with Stiff Detonation

  • Bin Yu (a1), Linying Li (a1), Bin Zhang (a1) and Jianhang Wang (a1)
Abstract

Incorrect propagation speed of discontinuities may occur by straightforward application of standard dissipative schemes for problems that contain stiff source terms for underresolved grids even for time steps within the CFL condition. By examining the dissipative discretized counterpart of the Euler equations for a detonation problem that consists of a single reaction, detailed analysis on the spurious wave pattern is presented employing the fractional step method, which utilizes the Strang splitting. With the help of physical arguments, a threshold values method (TVM), which can be extended to more complicated stiff problems, is developed to eliminate the wrong shock speed phenomena. Several single reaction detonations as well as multispecies and multi-reaction detonation test cases with strong stiffness are examined to illustrate the performance of the TVM approach.

Copyright
Corresponding author
*Corresponding author. Email address: zhangbin1983@sjtu.edu.cn (B. Zhang)
Footnotes
Hide All

Communicated by Chi-Wang Shu

Footnotes
References
Hide All
[1] Colella, P., Majda, A. and Roytburd, V., Theoretical and numerical structure for reacting shock waves, SIAM J. Sci. Stat. Comput., 7(4) (1986), 10591080.
[2] 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(5) (1992), 12441260.
[3] Lafon, A. and Yee, H. C., Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations, part iii: the effects of nonlinear source terms in reaction-convection equations, Int. J. Comput. Fluid Dyn., 6(1) (1996), 136.
[4] Lafon, A. and Yee, H. C., Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations part IV: stability vs. methods of discretizing nonlinear source terms in reaction-convection equations, Int. J. Comput. Fluid Dyn., 6(2) (1996), 89123.
[5] LeVeque, R. J. and Yee, H. C., A study of numerical methods for hyperbolic conservation laws with stiff source terms, J. Comput. Phys., 86(1) (1990), 187210.
[6] Yee, H. C., Kotov, D. V., Wang, W. and Shu, C.-W., Spurious behavior of shock-capturing methods by the fractional step approach: Problems containing stiff source terms and discontinuities, J. Comput. Phys., 241 (2013), 266291.
[7] Yee, H. C., Kotov, D. V., Wang, W. and Shu, C.-W., Corrigendum to “Spurious behavior of shock-capturing methods by the fractional step approach: Problems containing stiff source terms and discontinuities” [J. Comput. Phys., 241 (2013), 266–291], J. Comput. Phys., 250(1) (2013), 703712.
[8] Zhang, B. and Wang, J.-H., A short note on the counter-intuitive spurious behaviors in stiff reacting flow, J. Comput. Phys., 291 (2015), 5259.
[9] Bihari, B. L. and Schwendeman, D., Multiresolution schemes for the reactive euler equations, J. Comput. Phys., 154(1) (1999), 197230.
[10] Jeltsch, R. and Klingenstein, P., Error estimators for the position of discontinuities in hyperbolic conservation laws with source terms which are solved using operator splitting, Comput. Visual. Sci., 1(4) (1999), 231249.
[11] Leveque, R. J. and Shyue, K.-M., One-dimensional front tracking based on high resolution wave propagation methods, SIAM J. Sci. Comput., 16(2) (1995), 348377.
[12] Nguyen, D., Gibou, F. and Fedkiw, R., A fully conservative ghost fluid method and stiff detonation waves, in: 12th Int. Detonation Symposium, San Diego, CA, 2002.
[13] Sun, Y. and Engquist, B., Heterogeneous multiscale methods for interface tracking of combustion fronts, Multiscale Modeling & Simulation, 5(2) (2006), 532563.
[14] Chorin, A. J., Random choice solution of hyperbolic systems, J. Comput. Phys., 22(4) (1976), 517533.
[15] Chorin, A. J., Random choice methods with applications to reacting gas flow, J. Comput. Phys., 25(3) (1977), 253272.
[16] Majda, A. and Roytburd, V., Numerical study of the mechanisms for initiation of reacting shock waves, SIAM J. Sci. Stat. Comput., 11(5) (1990), 950974.
[17] Helzel, C., Leveque, R. J. and Warnecke, G., A modified fractional step method for the accurate approximation of detonation waves, SIAM J. Sci. Comput., 22(4) (2000), 14891510.
[18] Bao, W. and Jin, S., The random projection method for hyperbolic conservation laws with stiff reaction terms, J. Comput. Phys., 163(1) (2000), 216248.
[19] Bao, W. and Jin, S., The random projection method for stiff detonation capturing, SIAM J. Sci. Comput., 23(3) (2001), 10001026.
[20] Bao, W. and Jin, S., The random projection method for stiff multispecies detonation capturing, J. Comput. Phys., 178(1) (2002), 3757.
[21] Tosatto, L. and Vigevano, L., Numerical solution of under-resolved detonations, J. Comput. Phys., 227(4) (2008), 23172343.
[22] 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(1) (2012), 190214.
[23] Wang, W., Shu, C.-W., Yee, H. C., Kotov, D. V. and Sjögreen, B., High order finite difference methods with subcell resolution for stiff multispecies discontinuity capturing, Commun. Comput. Phys., 17(02) (2015), 317336.
[24] Zhang, B., Liu, H., Chen, F. and Wang, J. H., The equilibrium state method for hyperbolic conservation laws with stiff reaction terms, J. Comput. Phys., 263 (2014), 151176.
[25] Ben-Artzi, M., The generalized riemann problem for reactive flows, J. Comput. Phys., 81(1) (1989), 70101.
[26] Berkenbosch, A., Kaasschieter, E. and Klein, R., Detonation capturing for stiff combustion chemistry, Combustion Theory Modelling, 2(3) (1998), 313348.
[27] Bourlioux, A., Majda, A. J. and Roytburd, V., Theoretical and numerical structure for unstable one-dimensional detonations, SIAM J. Appl. Math., 51(2) (1991), 303343.
[28] Hidalgo, A. and Dumbser, M., Ader schemes for nonlinear systems of stiff advection–diffusion–reaction equations, J. Sci. Comput., 48(1-3) (2011), 173189.
[29] Miniati, F. and Colella, P., A modified higher order godunovs scheme for stiff source conservative hydrodynamics, J. Comput. Phys., 224(2) (2007), 519538.
[30] Pember, R. B., Numerical methods for hyperbolic conservation laws with stiff relaxation I, spurious solutions, SIAM J. Appl. Math., 53(5) (1993), 12931330.
[31] Ton, V. T., Improved shock-capturing methods for multicomponent and reacting flows, J. Comput. Phys., 128(1) (1996), 237253.
[32] Anderson, J. D., Hypersonic and high temperature gas dynamics, AIAA, 2000.
[33] Strang, G., On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5(3) (1968), 506517.
[34] Kurganov, A. and Tadmor, E., New high-resolution central schemes for nonlinear conservation laws and convection–diffusion equations, J. Comput. Phys., 160(1) (2000), 241282.
[35] Liou, M.-S. and Steffen, C. J., A new flux splitting scheme, J. Comput. Phys., 107(1) (1993), 2339.
[36] Liou, M.-S., A sequel to ausm: Ausm+, J. Comput. Phys., 129(2) (1996), 364382.
[37] Butcher, J. C., The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods, Wiley-Interscience, 1987.
[38] Kotov, D. V., Yee, H. C., Panesi, M., Prabhu, D. K. and Wray, A. A., Computational challenges for simulations related to the NASA electric arc shock tube (EAST) experiments, J. Comput. Phys., 269(1) (2014), 215233.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Communications in Computational Physics
  • ISSN: 1815-2406
  • EISSN: 1991-7120
  • URL: /core/journals/communications-in-computational-physics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed