Numerical simulations can be the key to the thorough understanding of the multi-dimensional nature of transient detonation waves. But the accurate approximation of realistic detonations is extremely demanding, because a wide range of different scales needs to be resolved. In this paper, we summarize our successful efforts in simulating multi-dimensional detonations with detailed and highly stiff chemical kinetics on recent parallel machines with distributed memory, especially on clusters of standard personal computers. We explain the design of AMROC, a freely available dimension-independent mesh adaptation framework for time-explicit Cartesian finite volume methods on distributed memory machines, and discuss the locality-preserving rigorous domain decomposition technique it employs. The framework provides a generic implementation of the blockstructured adaptive mesh refinement algorithm after Berger and Collela designed especially for the solution of hyperbolic fluid flow problems on logically rectangular grids. The ghost fluid approach is integrated into the refinement algorithm to allow for embedded non-Cartesian boundaries represented implicitly by additional level-set variables. Two- and three-dimensional simulations of regular cellular detonation structure in purely Cartesian geometry and a two-dimensional detonation propagating through a smooth 60 degree pipe bend are presented. Briefly, the employed upwind scheme and the treatment of the non-equilibrium reaction terms are sketched.


Detonation Wave Triple Point Slip Line Incident Shock Detonation Front 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bell, J., Berger, M., Saltzman, J., Welcome, M.: Three-dimensional adaptive mesh refinement for hyp. conservation laws. SIAM J. Sci. Comp. 15(1), 127–138 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Berger, M., Colella, P.: Local adaptive mesh refinement for shock hydrodynamics. J. Comput. Phys. 82, 64–84 (1988)CrossRefGoogle Scholar
  3. 3.
    Courant, R., Friedrichs, K.O.: Supersonic flow and shock waves. Applied mathematical sciences 21 (1976)Google Scholar
  4. 4.
    Deiterding, R.: Parallel adaptive simulation of multi-dimensional detonation structures (PhD thesis, Brandenburgische Technische Universität Cottbus (2003)Google Scholar
  5. 5.
    Deiterding, R.: AMROC - Blockstructured Adaptive Mesh Refinement in Object-oriented C++ (2005), Available at
  6. 6.
    Einfeldt, B., Munz, C.D., Roe, P.L., Sjögreen, B.: On Godunov-type methods near low densities. J. Comput. Phys. 92, 273–295 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Fedkiw, R.P., Aslam, T., Merriman, B., Osher, S.: A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J. Comput. Phys. 152, 457–492 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Grossmann, B., Cinella, P.: Flux-split algorithms for flows with non-equilibrium chemistry and vibrational relaxation. J. Comput. Phys. 88, 131–168 (1990)CrossRefGoogle Scholar
  9. 9.
    Hu, X.Y., Khoo, B.C., Zhang, D.L., Jiang, Z.L.: The cellular structure of a two-dimensional H2/O2/Ar detonation wave. Combustion Theory and Modelling 8, 339–359 (2004)CrossRefGoogle Scholar
  10. 10.
    Kaps, P., Rentrop, P.: Generalized Runge-Kutta methods of order four with stepsize control for stiff ordinary differential equations. Num. Math. 33, 55–68 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kee, R.J., Rupley, F.M., Miller, J.A.: Chemkin-II: A Fortran chemical kinetics package for the analysis of gas-phase chemical kinetics (SAND89-8009, Sandia National Laboratories, Livermore (1989)Google Scholar
  12. 12.
    Larrouturou, B.: How to preserve the mass fractions positivity when computing compressible multi-component flows. J. Comput. Phys. 95, 59–84 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    LeVeque, R.J.: Wave propagation algorithms for multidimensional hyperbolic systems. J. Comput. Phys. 131(2), 327–353 (1997)zbMATHCrossRefGoogle Scholar
  14. 14.
    Oran, E.S., Weber, J.W., Stefaniw, E.I., Lefebvre, M.H., Anderson, J.D.: A numerical study of a two-dimensional H2-O2-Ar detonation using a detailed chemical reaction model. J. Combust. Flame 113, 147–163 (1998)CrossRefGoogle Scholar
  15. 15.
    Osher, S., Fedkiw, R.: Level set methods and dynamic implicit surfaces. Applied Mathematical Science 153 (2003)Google Scholar
  16. 16.
    Parashar, M., Browne, J.C.: On partitioning dynamic adaptive grid hierarchies. In: Proc. of 29th Annual Hawaii Int. Conf. on System Sciences (1996)Google Scholar
  17. 17.
    Parashar, M., Browne, J.C.: System engineering for high performance computing software: The HDDA/DAGH infrastructure for implementation of parallel structured adaptive mesh refinement. In: Structured Adaptive Mesh Refinement Grid Methods. Mathematics and its Applications. Springer, Heidelberg (1997)Google Scholar
  18. 18.
    Quirk, J.J.: Godunov-type schemes applied to detonation flows. In: Buckmaster, J. (ed.) Combustion in high-speed flows, Proc. Workshop on Combustion, Hampton, October 12-14, pp. 575–596. Kluwer Acad. Publ., Dordrecht (1992)Google Scholar
  19. 19.
    Rendleman, C.A., Beckner, V.E., Lijewski, M., Crutchfield, W., Bell, J.B.: Parallelization of structured, hierarchical adaptive mesh refinement algorithms. Computing and Visualization in Science 3 (2000)Google Scholar
  20. 20.
    Sanders, R., Morano, E., Druguett, M.-C.: Multidimensional dissipation for upwind schemes: Stability and applications to gas dynamics. J. Comput. Phys. 145, 511–537 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Strehlow, R.A.: Gas phase detonations: Recent developments. J. Combust. Flame 12(2), 81–101 (1968)CrossRefGoogle Scholar
  22. 22.
    Toro, E.F.: Riemann solvers and numerical methods for fluid dynamics. Springer, Heidelberg (1999)zbMATHGoogle Scholar
  23. 23.
    Tsuboi, N., Katoh, S., Hayashi, A.K.: Three-dimensional numerical simulation for hydrogen/air detonation: Rectangular and diagonal structures. Proc. Combustion Institute 29, 2783–2788 (2003)CrossRefGoogle Scholar
  24. 24.
    Westbrook, C.K.: Chemical kinetics of hydrocarbon oxidation in gaseous detonations. J. Combust. Flame 46, 191–210 (1982)CrossRefGoogle Scholar
  25. 25.
    Williams, D.N., Bauwens, L., Oran, E.S.: Detailed structure and propagation of three-dimensional detonations. Proc. Combustion Institute 26, 2991–2998 (1997)CrossRefGoogle Scholar
  26. 26.
    Williams, F.A.: Combustion theory. Addison-Wesley, Reading (1985)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Ralf Deiterding
    • 1
  1. 1.California Institute of TechnologyPasadena

Personalised recommendations