Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-27T10:03:07.034Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Shock waves and blast waves Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  03 May 2017

Bin Yu ,
Linying Li ,
Bin Zhang  and
Jianhang Wang
Bin Yu
Affiliation:
School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China
Linying Li
Affiliation:
School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China
Bin Zhang*
Affiliation:
School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China
Jianhang Wang
Affiliation:
School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China
*
*Corresponding author. Email address:zhangbin1983@sjtu.edu.cn (B. Zhang)
Get access

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.

Keywords

Detonationspurious behaviorreactive Euler equationsthreshold values methodstiffness

MSC classification

Secondary: 35Q31: Euler equations 76L05: Shock waves and blast waves
Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 1 , July 2017 , pp. 259 - 284
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Communicated by Chi-Wang Shu

References

[1] Colella, P., Majda, A. and Roytburd, V., Theoretical and numerical structure for reacting shock waves, SIAM J. Sci. Stat. Comput., 7(4) (1986), 1059–1080.CrossRefGoogle Scholar
[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), 1244–1260.CrossRefGoogle Scholar
[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), 1–36.CrossRefGoogle Scholar
[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), 89–123.CrossRefGoogle Scholar
[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), 187–210.CrossRefGoogle Scholar
[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), 266–291.CrossRefGoogle Scholar
[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), 703–712.CrossRefGoogle Scholar
[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), 52–59.CrossRefGoogle Scholar
[9] Bihari, B. L. and Schwendeman, D., Multiresolution schemes for the reactive euler equations, J. Comput. Phys., 154(1) (1999), 197–230.CrossRefGoogle Scholar
[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), 231–249.CrossRefGoogle Scholar
[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), 348–377.CrossRefGoogle Scholar
[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.Google Scholar
[13] Sun, Y. and Engquist, B., Heterogeneous multiscale methods for interface tracking of combustion fronts, Multiscale Modeling & Simulation, 5(2) (2006), 532–563.CrossRefGoogle Scholar
[14] Chorin, A. J., Random choice solution of hyperbolic systems, J. Comput. Phys., 22(4) (1976), 517–533.CrossRefGoogle Scholar
[15] Chorin, A. J., Random choice methods with applications to reacting gas flow, J. Comput. Phys., 25(3) (1977), 253–272.CrossRefGoogle Scholar
[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), 950–974.CrossRefGoogle Scholar
[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), 1489–1510.CrossRefGoogle Scholar
[18] Bao, W. and Jin, S., The random projection method for hyperbolic conservation laws with stiff reaction terms, J. Comput. Phys., 163(1) (2000), 216–248.CrossRefGoogle Scholar
[19] Bao, W. and Jin, S., The random projection method for stiff detonation capturing, SIAM J. Sci. Comput., 23(3) (2001), 1000–1026.CrossRefGoogle Scholar
[20] Bao, W. and Jin, S., The random projection method for stiff multispecies detonation capturing, J. Comput. Phys., 178(1) (2002), 37–57.CrossRefGoogle Scholar
[21] Tosatto, L. and Vigevano, L., Numerical solution of under-resolved detonations, J. Comput. Phys., 227(4) (2008), 2317–2343.CrossRefGoogle Scholar
[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), 190–214.CrossRefGoogle Scholar
[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), 317–336.CrossRefGoogle Scholar
[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), 151–176.CrossRefGoogle Scholar
[25] Ben-Artzi, M., The generalized riemann problem for reactive flows, J. Comput. Phys., 81(1) (1989), 70–101.CrossRefGoogle Scholar
[26] Berkenbosch, A., Kaasschieter, E. and Klein, R., Detonation capturing for stiff combustion chemistry, Combustion Theory Modelling, 2(3) (1998), 313–348.CrossRefGoogle Scholar
[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), 303–343.CrossRefGoogle Scholar
[28] Hidalgo, A. and Dumbser, M., Ader schemes for nonlinear systems of stiff advection–diffusion–reaction equations, J. Sci. Comput., 48(1-3) (2011), 173–189.CrossRefGoogle Scholar
[29] Miniati, F. and Colella, P., A modified higher order godunovs scheme for stiff source conservative hydrodynamics, J. Comput. Phys., 224(2) (2007), 519–538.CrossRefGoogle Scholar
[30] Pember, R. B., Numerical methods for hyperbolic conservation laws with stiff relaxation I, spurious solutions, SIAM J. Appl. Math., 53(5) (1993), 1293–1330.CrossRefGoogle Scholar
[31] Ton, V. T., Improved shock-capturing methods for multicomponent and reacting flows, J. Comput. Phys., 128(1) (1996), 237–253.CrossRefGoogle Scholar
[32] Anderson, J. D., Hypersonic and high temperature gas dynamics, AIAA, 2000.Google Scholar
[33] Strang, G., On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5(3) (1968), 506–517.CrossRefGoogle Scholar
[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), 241–282.CrossRefGoogle Scholar
[35] Liou, M.-S. and Steffen, C. J., A new flux splitting scheme, J. Comput. Phys., 107(1) (1993), 23–39.CrossRefGoogle Scholar
[36] Liou, M.-S., A sequel to ausm: Ausm+, J. Comput. Phys., 129(2) (1996), 364–382.CrossRefGoogle Scholar
[37] Butcher, J. C., The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods, Wiley-Interscience, 1987.Google Scholar
[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), 215–233.CrossRefGoogle Scholar