Skip to main content
SpringerLink
Log in

A Modified A Posteriori Subcell Limiter for High Order Flux Reconstruction Scheme for One-Dimensional Detonation Simulation

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this paper, a modified a posteriori limiter is developed for high order flux reconstruction (FR) scheme for the numerical simulation of detonation problems. In this limiting procedure, the unlimited FR solution at the new time step will be checked first by using some detection criteria, then the solution in the troubled cells are recomputed with a robust subcell finite volume (FV) scheme. The detection criteria for identifying troubled cells consist of the physical admissibility (e.g., positivity of density and pressure) and numerical admissibility (e.g., non-oscillating). We modify the detection criteria by using the KXRCF shock detector prior to the relaxed discrete maximum principle. This can track the troubled cells near strong shocks consecutively so as to improve the steady state convergence and can reduce the number of overly marked troubled cells. The subcell correction procedure endows the high order FR scheme the capability to capture discontinuities inside a cell without generating spurious oscillations. A series of one-dimensional numerical tests are carried out to assess the effectiveness of the proposed limiter. In particular, one-dimensional detonation wave problems with the overdriven factor f = 1.8–1.3 are calculated using third to sixth order accurate FR schemes in conjunction with the first order Godunov or second order TVD subcell FV scheme. It is shown that the FR schemes with the present a posteriori limiter can compute strong detonation waves robustly, and the third order FR scheme with the second order TVD subcell FV limiter has better resolution of detonation waves compared with the fifth order WENO-Z scheme under same degree of freedoms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Algorithm 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19

Similar content being viewed by others

Data Availability

Enquiries about data availability should be directed to the authors.

References

  1. Zhu, H.Q., Gao, Z.: An h-adaptive RKDG method with troubled-cell indicator for one-dimensional detonation wave simulations. Adv. Comput. Math. 42, 1081–1102 (2016). https://doi.org/10.1007/s10444-016-9454-3

    Article  MathSciNet  MATH  Google Scholar 

  2. Henshaw, W.D., Schwendeman, D.W.: An adaptive numerical scheme for high-speed reactive flow on overlapping grids. J. Comput. Phys. 191(2), 420–447 (2003). https://doi.org/10.1016/S0021-9991(03)00323-1

    Article  MathSciNet  MATH  Google Scholar 

  3. Hu, G.H.: A numerical study of 2D detonation waves with adaptive finite volume methods on unstructured grids. J. Comput. Phys. 331, 297–311 (2017). https://doi.org/10.1016/j.jcp.2016.11.041

    Article  MathSciNet  MATH  Google Scholar 

  4. Henrick, A.K., Aslam, T.D., Powers, J.M.: Simulations of pulsating one-dimensional detonations with true fifth order accuracy. J. Comput. Phys. 213(1), 311–329 (2006). https://doi.org/10.1016/j.jcp.2005.08.013

    Article  MathSciNet  MATH  Google Scholar 

  5. Gao, Z., Don, W.S., Li, Z.Q.: High order weighted essentially non-oscillation schemes for one-dimensional detonation wave simulations. J. Comput. Math. 29(6), 623–638 (2011). https://doi.org/10.4208/jcm.1110-m11si02

    Article  MathSciNet  MATH  Google Scholar 

  6. Wang, C., Zhang, X., Shu, C.-W., Ning, J.G.: Robust high order discontinuous Galerkin schemes for two-dimensional gaseous detonations. J. Comput. Phys. 231(2), 653–665 (2012). https://doi.org/10.1016/j.jcp.2011.10.002

    Article  MathSciNet  MATH  Google Scholar 

  7. Huynh, H.T.: A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods. In: 18th AIAA Computational Fluid Dynamics Conference, AIAA 2007-4079 (2007)

  8. Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer, Berlin (2007)

    MATH  Google Scholar 

  9. Kopriva, D.A., Kolias, J.H.: A conservative staggered-grid Chebyshev multidomain method for compressible flows. J. Comput. Phys. 125(1), 244–261 (1996). https://doi.org/10.1006/jcph.1996.0091

    Article  MathSciNet  MATH  Google Scholar 

  10. Liu, Y., Vinokur, M., Wang, Z.J.: Spectral difference method for unstructured grids I: basic formulation. J. Comput. Phys. 216(2), 780–801 (2006). https://doi.org/10.1016/j.jcp.2006.01.024

    Article  MathSciNet  MATH  Google Scholar 

  11. López-Morales, M.R., Bull, J., Grabill, J., et al.: Verification and validation of HiFiLES: a high-order LES unstructured solver on multi-GPU platforms. In: 32nd AIAA Applied Aerodynamics Conference, AIAA 2014-3168 (2014)

  12. Witherden, F.D., Farrington, A.M., Vincent, P.E.: PyFR: An open source framework for solving advection-diffusion type problems on streaming architectures using the flux reconstruction approach. Comput. Phys. Commun. 185(11), 3028–3040 (2014). https://doi.org/10.1016/j.cpc.2014.07.011

    Article  MATH  Google Scholar 

  13. Romero, J., Crabill, J., Watkins, J.E., Witherden, F.D., Jameson, A.: ZEFR: A GPU-accelerated high-order solver for compressible viscous flows using the flux reconstruction method. Comput. Phys. Commun. 250, 107169 (2020). https://doi.org/10.1016/j.cpc.2020.107169

    Article  MATH  Google Scholar 

  14. Jameson, A.: A proof of the stability of the spectral difference method for all orders of accuracy. J. Sci. Comput. 45, 348–358 (2010). https://doi.org/10.1007/s10915-009-9339-4

    Article  MathSciNet  MATH  Google Scholar 

  15. Vincent, P.E., Castonguay, P., Jameson, A.: A new class of high-order energy stable flux reconstruction schemes. J. Sci. Comput. 47, 50–72 (2011). https://doi.org/10.1007/s10915-010-9420-z

    Article  MathSciNet  MATH  Google Scholar 

  16. Wang, Z.J., Gao, H.: A unifying lifting collocation penalty formulation including the discontinuous Galerkin, spectral volume/difference methods for conservation laws on mixed grids. J. Comput. Phys. 228(21), 8161–8186 (2009). https://doi.org/10.1016/j.jcp.2009.07.036

    Article  MathSciNet  MATH  Google Scholar 

  17. Yu, M.L., Wang, Z.J.: On the connection between the correction and weighting functions in the correction procedure via reconstruction method. J. Sci. Comput. 54, 227–244 (2013). https://doi.org/10.1007/s10915-012-9618-3

    Article  MathSciNet  MATH  Google Scholar 

  18. Huynh, H.T., Wang, Z.J., Vincent, P.E.: High-order methods for computational fluid dynamics: a brief review of compact differential formulations on unstructured grids. Comput. Fluids 98, 209–220 (2014). https://doi.org/10.1016/j.compfluid.2013.12.007

    Article  MathSciNet  MATH  Google Scholar 

  19. Godunov, S.K., Bohachevsky, I.: Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics. Matematičeskij sbornik 47(3), 271–306 (1959)

    Google Scholar 

  20. Castonguay, P., Williams, D.M., Vincent, P.E., Jameson, A.: Energy stable flux reconstruction schemes for advection-diffusion problems. Comput. Methods Appl. Mech. Eng. 267, 400–417 (2013). https://doi.org/10.1016/j.cma.2013.08.012

    Article  MathSciNet  MATH  Google Scholar 

  21. Harten, A.: High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49(3), 357–393 (1983). https://doi.org/10.1016/0021-9991(83)90136-5

    Article  MathSciNet  MATH  Google Scholar 

  22. Sweby, P.K.: High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal. 21(5), 995–1011 (1984). https://doi.org/10.1137/0721062

    Article  MathSciNet  MATH  Google Scholar 

  23. Cockburn, B., Shu, C.-W.: The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys. 141(2), 199–224 (1998). https://doi.org/10.1006/jcph.1998.5892

    Article  MathSciNet  MATH  Google Scholar 

  24. Park, J.S., Yoon, S.-H., Kim, C.: Multi-dimensional limiting process for hyperbolic conservation laws on unstructured grids. J. Comput. Phys. 229(3), 788–812 (2010). https://doi.org/10.1016/j.jcp.2009.10.011

    Article  MathSciNet  MATH  Google Scholar 

  25. Park, J.S., Kim, C.: Hierarchical multi-dimensional limiting strategy for correction procedure via reconstruction. J. Comput. Phys. 308, 57–80 (2016). https://doi.org/10.1016/j.jcp.2015.12.020

    Article  MathSciNet  MATH  Google Scholar 

  26. Biswas, R., Devine, K.D., Flaherty, J.E.: Parallel, adaptive finite element methods for conservation laws. Appl. Numer. Math. 14(1), 255–283 (1994). https://doi.org/10.1016/0168-9274(94)90029-9

    Article  MathSciNet  MATH  Google Scholar 

  27. Burbeau, A., Sagaut, P., Bruneau, C.-H.: A problem-independent limiter for high-order Runge-Kutta discontinuous Galerkin methods. J. Comput. Phys. 169(1), 111–150 (2001). https://doi.org/10.1006/jcph.2001.6718

    Article  MathSciNet  MATH  Google Scholar 

  28. Krivodonova, L.: Limiters for high-order discontinuous Galerkin methods. J. Comput. Phys. 226(1), 879–896 (2007). https://doi.org/10.1016/j.jcp.2007.05.011

    Article  MathSciNet  MATH  Google Scholar 

  29. Yang, M., Wang, Z.J.: A parameter-free generalized moment limiter for high-order methods on unstructured grids. Adv. Appl. Math. Mech. 1(4), 451–480 (2009). https://doi.org/10.4208/aamm.09-m0913

    Article  MathSciNet  Google Scholar 

  30. Qiu, J., Shu, C.W.: Runge-Kutta discontinuous Galerkin method using WENO limiters. SIAM J. Sci. Comput. 26(3), 907–929 (2005). https://doi.org/10.1137/S1064827503425298

    Article  MathSciNet  MATH  Google Scholar 

  31. Zhu, J., Qiu, J.X., Shu, C.-W., Dumbser, M.: Runge-Kutta discontinuous Galerkin method using WENO limiters II: unstructured meshes. J. Comput. Phys. 227(9), 4330–4353 (2008). https://doi.org/10.1016/j.jcp.2007.12.024

    Article  MathSciNet  MATH  Google Scholar 

  32. Qiu, J.X., Shu, C.-W.: Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method: One-dimensional case. J. Comput. Phys. 193(1), 115–135 (2004). https://doi.org/10.1016/j.jcp.2003.07.026

    Article  MathSciNet  MATH  Google Scholar 

  33. Qiu, J.X., Shu, C.-W.: Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method II: Two dimensional case. Comput. Fluids 34(6), 642–663 (2005). https://doi.org/10.1016/j.compfluid.2004.05.005

    Article  MathSciNet  MATH  Google Scholar 

  34. Zhong, X., Shu, C.-W.: A simple weighted essentially nonoscillatory limiter for Runge-Kutta discontinuous Galerkin methods. J. Comput. Phys. 232(1), 397–415 (2013). https://doi.org/10.1016/j.jcp.2012.08.028

    Article  MathSciNet  Google Scholar 

  35. Zhang, X., Shu, C.-W.: On maximum-principle-satisfying high order schemes for scalar conservation laws. J. Comput. Phys. 229(9), 3091–3120 (2010). https://doi.org/10.1016/j.jcp.2009.12.030

    Article  MathSciNet  MATH  Google Scholar 

  36. Zhang, X., Shu, C.-W.: On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes. J. Comput. Phys. 229(23), 8918–8934 (2010). https://doi.org/10.1016/j.jcp.2010.08.016

    Article  MathSciNet  MATH  Google Scholar 

  37. Hu, X.Y., Adams, N.A., Shu, C.-W.: Positivity-preserving method for high-order conservative schemes solving compressible Euler equations. J. Comput. Phys. 242, 169–180 (2013). https://doi.org/10.1016/j.jcp.2013.01.024

    Article  MathSciNet  MATH  Google Scholar 

  38. Clain, S., Diot, S., Loubére, R.: A high-order finite volume method for systems of conservation laws-multi-dimensional optimal order detection (MOOD). J. Comput. Phys. 230(10), 4028–4050 (2011). https://doi.org/10.1016/j.jcp.2011.02.026

    Article  MathSciNet  MATH  Google Scholar 

  39. Diot, S., Clain, S., Loubére, R.: Improved detection criteria for the multi-dimensional optimal order detection (MOOD) on unstructured meshes with very high-order polynomials. Comput. Fluids 64, 43–63 (2012). https://doi.org/10.1016/j.compfluid.2012.05.004

    Article  MathSciNet  MATH  Google Scholar 

  40. Dumbser, M., Zanotti, O., Loubére, R., Diot, S.: A posteriori subcell limiting of the discontinuous Galerkin finite element method for hyperbolic conservation laws. J. Comput. Phys. 278, 47–75 (2014). https://doi.org/10.1016/j.jcp.2014.08.009

    Article  MathSciNet  MATH  Google Scholar 

  41. Sonntag, M., Munz, C.-D.: Shock capturing for discontinuous Galerkin methods using finite volume subcells. In: Fuhrmann, J., Ohlberger, M., Rohde, C. (eds.) Finite Volumes for Complex Applications VII-Elliptic, Parabolic and Hyperbolic Problems, pp. 945–953. Springer, Cham (2014)

    Chapter  MATH  Google Scholar 

  42. Vilar, F.: A posteriori correction of high-order discontinuous Galerkin scheme through subcell finite volume formulation and flux reconstruction. J. Comput. Phys. 387, 245–279 (2019). https://doi.org/10.1016/j.jcp.2018.10.050

    Article  MathSciNet  MATH  Google Scholar 

  43. Li, Y., Wang, Z.J.: Recent progress in a convergent and accuracy preserving limiter for the FR/CPR method. AIAA 2017-0756 (2017). https://doi.org/10.2514/6.2017-0756

  44. Krivodonova, L., Xin, J., Remacle, J.-F., Chevaugeon, N., Flaherty, J.E.: Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws. Appl. Numer. Math. 48(3), 323–338 (2004). https://doi.org/10.1016/j.apnum.2003.11.002

    Article  MathSciNet  MATH  Google Scholar 

  45. Vincent, P.E., Castonguay, P., Jameson, A., Huynh, H.T.: Insights from von Neumann analysis of high-order flux reconstruction schemes. J. Comput. Phys. 230(22), 8134–8154 (2011). https://doi.org/10.1016/j.jcp.2011.07.013

    Article  MathSciNet  MATH  Google Scholar 

  46. Jameson, A., Vincent, P.E., Castonguay, P.: On the non-linear stability of flux reconstruction schemes. J. Sci. Comput. 50, 434–445 (2012). https://doi.org/10.1007/s10915-011-9490-6

    Article  MathSciNet  MATH  Google Scholar 

  47. Witherden, F.D., Vincent, P.E.: On nodal point sets for flux reconstruction. J. Comput. Appl. Math. 381, 113014 (2021). https://doi.org/10.1016/j.cam.2020.113014

    Article  MathSciNet  MATH  Google Scholar 

  48. Toro, E.: Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction. Springer, USA (2009). https://doi.org/10.1007/b79761

    Book  MATH  Google Scholar 

  49. Pan, J.H., Ren, Y.X.: High order sub-cell finite volume schemes for solving hyperbolic conservation laws I: basic formulation and one-dimensional analysis. Sci. China Phys. Mech. Astron. 60(8), 084711 (2017). https://doi.org/10.1007/s11433-017-9033-9

    Article  Google Scholar 

  50. Runge, C.: Über empirische funktionen und die interpolation zwischen äquidistanten ordinaten. Z. Angew. Math. Phys. 46, 224–243 (1901)

    MATH  Google Scholar 

  51. Godunov, S.K.: A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics. Math. Sbornik Novaya Seriya 47(3), 271–306 (1959)

    MathSciNet  MATH  Google Scholar 

  52. van Leer, B.: Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method. J. Comput. Phys. 32(1), 101–136 (1979). https://doi.org/10.1016/0021-9991(79)90145-1

    Article  MATH  Google Scholar 

  53. Harten, A., Osher, S.: Uniformly high-order accurate nonoscillatory schemes I. SIAM J. Numer. Anal. 24(2), 279–309 (1987). https://doi.org/10.1137/0724022

    Article  MathSciNet  MATH  Google Scholar 

  54. Gao, Z., Don, W.S., Li, Z.Q.: High order weighted essentially non-oscillation schemes for two-dimensional detonation wave simulations. J. Sci. Comput. 53, 80–101 (2012). https://doi.org/10.1007/s10915-011-9569-0

    Article  MathSciNet  MATH  Google Scholar 

  55. Zhang, Z.C., Yu, S.-T., He, H., Chang, S.-C.: Direct calculations of two-and three-dimensional detonations by an extended CE/SE method. AIAA 2001-0476 (2001). https://doi.org/10.2514/6.2001-476

  56. Wang, B., He, H., Yu, S.-T.: Direct calculation of wave implosion for detonation initiation. AIAA J. 43(10), 2157–2169 (2005). https://doi.org/10.2514/1.11887

    Article  Google Scholar 

  57. Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77(2), 439–471 (1988). https://doi.org/10.1016/0021-9991(88)90177-5

    Article  MathSciNet  MATH  Google Scholar 

  58. Sod, G.A.: A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J. Comput. Phys. 27(1), 1–31 (1978). https://doi.org/10.1016/0021-9991(78)90023-2

    Article  MathSciNet  MATH  Google Scholar 

  59. Einfeldt, B., Munz, C.D., Roe, P.L., Sjögreen, B.: On Godunov-type methods near low densities. J. Comput. Phys. 92(2), 273–295 (1991). https://doi.org/10.1016/0021-9991(91)90211-3

    Article  MathSciNet  MATH  Google Scholar 

  60. Woodward, P., Colella, P.: The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54(1), 115–173 (1984). https://doi.org/10.1016/0021-9991(84)90142-6

    Article  MathSciNet  MATH  Google Scholar 

  61. Bourlioux, A., Majda, A.J., Roytburd, V.: Theoretical and numerical structure for unstable one-dimensional detonations. SIAM J. Appl. Math. 51(2), 303–343 (1991). https://doi.org/10.1137/0151016

    Article  MathSciNet  MATH  Google Scholar 

  62. Di, Y.N., Hu, G.H., Li, R., Yang, F.: On accurately resolving detonation dynamics by adaptive finite volume method on unstructured grids. Commun. Comput. Phys. 29(2), 445–471 (2020). https://doi.org/10.4208/cicp.OA-2020-0028

    Article  MathSciNet  MATH  Google Scholar 

  63. Deiterding, R.: Parallel adaptive simulation of multi-dimensional detonation structures. PhD thesis, Brandenburgische Technische Universitat Cottbus (2003). https://eprints.soton.ac.uk/380602/

  64. Karagozian, P., Hwang, P., Fedkiw, R., Merriman, B., Karagozian, A., Osher, S.: Numerical resolution of pulsating detonation waves. Combustion Theory and Modelling 4 (1970). https://doi.org/10.1088/1364-7830/4/3/301

  65. Jiang, Y., Shu, C.-W., Zhang, M.P.: An alternative formulation of finite difference weighted ENO schemes with Lax-Wendroff time discretization for conservation laws. SIAM J. Sci. Comput. 35, 1137–1160 (2013). https://doi.org/10.1137/120889885

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

This work is supported by Natural Science Foundation of China (Grant Nos. 91852116, 12071470, 12161141017). The computations are carried out on the high performance computer of the State Key Laboratory of Scientific and Engineering Computing, Chinese Academy of Sciences.

Author information

Authors and Affiliations

  1. Institute of Applied Physics and Computational Mathematics, China Academy of Engineering Physics, Beijing, 100094, China

    Shiwei Liu

  2. ICMSEC and LSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Li Yuan

  3. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China

    Shiwei Liu & Li Yuan

Authors
  1. Shiwei Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Li Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shiwei Liu.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical Approval and consent to participate

Not applicable.

Consent for Publication

Not applicable.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, S., Yuan, L. A Modified A Posteriori Subcell Limiter for High Order Flux Reconstruction Scheme for One-Dimensional Detonation Simulation. J Sci Comput 97, 31 (2023). https://doi.org/10.1007/s10915-023-02347-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-023-02347-7

Keywords

Search

Navigation