Shock Waves pp 1131-1136 | Cite as

A robust and simple upwind scheme: a way to resolve contact discontinuities and suppress the carbuncle instability

  • M. Sun
  • K. Takayama
Conference paper


A new method is proposed to split the flux vector of the Euler equations by introducing two artificial wave speeds. The flux vector is split to two simple flux vectors. One flux vector comes with unidirectional eigenvalues, so that is can be easily solved by one-side differencing. Another flux vector becomes a system of two waves and one, two or three stationary discontinuities depending on the dimension of the Euler equations. Numerical flux function for multi-dimensional Euler equations is formulated for any grid system, structured or unstructured. A remarkable simplicity of the scheme is that it successfully achieves one-sided approximation for all waves without recourse to any matrix operation. Moreover, its accuracy is comparable with the exact Riemann solver. For 1-D Euler equations, the scheme actually surpasses the exact solver in avoiding expansion shocks without any additional entropy fix. The scheme can exactly resolve stationary 1-D contact discontinuities, and it avoids the carbuncle problem in multi-dimensional computations. This paper is a guide to our recent work [3]


Euler Equation Wave Speed Contact Discontinuity Flux Vector Riemann Solver 
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.
    A. Harten, P.D. Lax, B. van Leer: On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Review 25, 35 (1983)MathSciNetCrossRefGoogle Scholar
  2. 2.
    M. Pandolfi, D. D’Ambrosio: Numerical instabilities in upwind methods: analysis and cures for the “Carbuncle” phenomenon. J. of Comput. Phys 166, 271 (2001)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    M. Sun, K. Takayama: An artificially-upstream flux vector splitting method for the Euler equations. J. of Comput. Phys (to be published)Google Scholar
  4. 4.
    E.F. Toro, M. Spruce, W. Speares: Restoration of the contact surface in the HLL-Riemann solver. Shock Waves 4, 25 (1994)ADSCrossRefGoogle Scholar

Copyright information

© Tsinghua University Press and Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • M. Sun
    • 1
  • K. Takayama
    • 1
  1. 1.Shock Wave Research Center, Institute of Fluid ScienceTohoku UniversitySendaiJapan

Personalised recommendations