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International Journal of Nonlinear Sciences and Numerical Simulation

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Volume 17, Issue 1 (Feb 2016)


Healing of the Carbuncle Phenomenon for AUSMDV Scheme on Triangular Grids

Sutthisak Phongthanapanich
  • Corresponding author
  • Department of Mechanical Engineering Technology, College of Industrial Technology, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand
  • Email:
Published Online: 2016-01-09 | DOI: https://doi.org/10.1515/ijnsns-2015-0008


The carbuncle phenomenon, commonly occurring in solutions of compressible Euler equations, is a numerical instability associated with shock-induced anomalies. It is associated with several shock-capturing finite-volume methods designed to preserve the contact discontinuities. Due to the lack of theoretical knowledge of the carbuncle phenomenon, it is not known which numerical scheme is affected or under what circumstances that the phenomenon occur. The objective of this article is to study the numerical instability of advection upstream splitting method (AUSM) family schemes so called the AUSMD, AUSMV and AUSMDV schemes in two-dimensional structured triangular grids by examining the shock-induced anomalies produced by these original schemes in different test cases. A multidimensional dissipation technique is proposed for these schemes. The evolution of perturbations is also studied by means of a linearized discrete analysis to the odd–even decoupling problem. The recursive equations show that the AUSMDV-family schemes with the dissipation technique are less sensitive to these anomalies than the original schemes. Finally, the dissipation technique is extended to the second-order schemes and tested by several test cases.

Keywords: AUSMDV scheme; carbuncle phenomenon; Euler equations; finite volume method; upwind method

PACS® (2010).: 02.60.Cb; 52.35.Tc; 47.11.Df; 47.20.Cq; 47.40.Ki


  • [1] S. K. Godunov, A difference method for the numerical calculation of discontinuous solutions of hydrodynamic equations, Mat. Sbornik 42 (1959), 271–306.

  • [2] J. L. Steger and R. F. Warming, Flux vector-splitting of the inviscid gas dynamic equations with application to finite-difference methods, J. Comput. Phys. 40 (1981), 263–293.

  • [3] B. Van Leer, Flux vector splitting for the Euler equation, Lect. Notes Phys. 170 (1982), 507–512.

  • [4] P. L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys. 43 (1981), 357–372.

  • [5] E. F. Toro, M. Spruce and W. Speares, Restoration of the contact surface in the HLL-Riemann solver, Shock Waves 4 (1994), 25–34.

  • [6] J. J. Quirk, Contribution to the great Riemann solver debate, Int. J. Numer. Meth. Fluids 18 (1994), 555–574.

  • [7] S. Phongthanapanich and P. Dechaumphai, Modified H-correction entropy fix for Roe’s flux-difference splitting scheme with mesh adaptation, Trans. Can. Soc. Mech. Eng. 28 (2004), 531–549.

  • [8] S. Phongthanapanich and P. Dechaumphai, Multidimensional dissipation technique for Roe’s flux-difference splitting scheme on triangular meshes, Int. J. Nonlinear Sci. Numer. Simul. 7 (2006), 251–256.

  • [9] S. Phongthanapanich and P. Dechaumphai, Healing of shock instability for Roe’s flux-difference splitting scheme on triangular meshes, Int. J. Numer. Meth. Fluids 59 (2009), 559–575.

  • [10] M. V. C. Ramalho, J. H. A. Azevedo and J. L. F. Azevedo, Further investigation into the origin of the carbuncle phenomenon in aerodynamic simulations, AIAA 2011–1184, In 49th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, Orlando, Florida, 2011.

  • [11] J.-Ch. Robinet, J. Gressier, G. Casalis and J. M. Moschetta, Shock wave instability and the carbuncle phenomenon: same intrinsic origin? J. Fluid Mech. 417 (2000), 237–263.

  • [12] M. Dumbser, J. M. Moschetta and J. Gressier, A matrix stability analysis of the carbuncle phenomenon, J. Comput. Phys. 197 (2004), 647–670.

  • [13] R. Sanders, E. Morano and M. C. Druguet, Multidimensional dissipation for upwind schemes: stability and applications to gas dynamics, J. Comput. Phys. 145 (1998), 511–537.

  • [14] M. Pandolfi and D. D‘Ambrosio, Numerical instabilities in upwind methods: analysis and cures for the “Carbuncle” phenomenon, J. Comput. Phys. 166 (2001), 271–301.

  • [15] M. S. Liou and C. J. Steffen, A new flux splitting scheme, J. Comput. Phys. 107 (1993), 23–39.

  • [16] M. S. Liou, A sequel to AUSM: AUSM+, J. Comput. Phys. 129 (1996), 364–382.

  • [17] Y. Wada and M. S. Liou, An accurate and robust flux splitting scheme for shock and contact discontinuities, SIAM J. Sci. Comput. 18 (1997), 633–657.

  • [18] K. H. Kim, J. H. Lee and O. H. Rho, An improvement of AUSM schemes by introducing the pressure-based weight functions, Comput. Fluids 27 (1998), 311–346.

  • [19] K. H. Kim, C.Kim and O. H. Rho, Methods for the accurate computations of hypersonic flows I. AUSMPW+ scheme, J. Comput. Phys. 174 (2001), 38–80.

  • [20] M. S. Liou, A sequel to AUSM, part II: AUSM+-up for all speeds, J. Comput. Phys. 214 (2006), 137–170.

  • [21] G. Sun, G. Wu and C. J. Liu, Numerical simulation of supersonic flow with shock wave using modified AUSM scheme, Int. J. Nonlinear Sci. Numer. Simul. 7 (2006), 329–332.

  • [22] K. Kitamura and E. Shima, Towards shock-stable and accurate hypersonic heating computations: a new pressure flux for AUSM-family schemes, J. Comput. Phys. 245 (2013), 62–83. [Web of Science]

  • [23] X. Jiang, C. Zhihua, F. Baochun and L. Hongzhi, Numerical simulation of blast flow fields induced by a high-speed projectile, Shock Waves 18 (2008), 205–212. [Web of Science]

  • [24] J. R. Shin and J. Y. Choi, DES study of base and base-bleed flows with dynamic formulation of DES constant, AIAA paper 2011–662, 49th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, Orlando, Florida, 2011.

  • [25] T. Nagao, M. Asahara, K. Hayashi, N. Tsuboi and E. Yamada, Numerical analysis of spinning detonation dependency on initial pressure using AUSMDV scheme, AIAA paper 2013–1177, 51st AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, Grapevine, Texas, 2013.

  • [26] R. J. Gollan and P. A. Jacobs, About the formulation, verification and validation of the hypersonic flow solver Eilmer, Int. J. Numer. Methods Fluids 73 (2013), 19–57. [Web of Science]

  • [27] V. Venkatakrishnan, Convergence to steady state solutions of the Euler equations on unstructured grids with limiters, J. Comput. Phys. 118 (1995), 120–130.

  • [28] C. W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys. 77 (1988), 439–471.

  • [29] S. Tu, A high order space-time Riemann-solver-free method for solving compressible Euler equations, AIAA paper 2009–1335, 47th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, Orlando, Florida, 2009.

  • [30] M. Arora and P. L. Roe, On postshock oscillations due to shock capturing schemes in unsteady flows, J. Comput. Phys. 130 (1997), 25–40.

  • [31] R. Fedkiw, X. D. Liu and S. Osher, A general technique for eliminating spurious oscillations in conservative schemes for multiphase and multispecies Euler equations, Int. J. Nonlinear Sci. Numer. Simul. 3 (2002), 99–106.

  • [32] R. Hillier, Computational of shock wave diffraction at a ninety degrees convex edge, Shock Waves 1 (1991), 89–98.

  • [33] K. Takayama and Z. Jiang, Shock wave reflection over wedges: a benchmark test for CFD and experiments, Shock Waves 7 (1997), 191–203.

About the article

Received: 2015-01-18

Accepted: 2015-12-02

Published Online: 2016-01-09

Published in Print: 2016-02-01

Citation Information: International Journal of Nonlinear Sciences and Numerical Simulation, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339, DOI: https://doi.org/10.1515/ijnsns-2015-0008. Export Citation

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