Journal of Engineering Mathematics

, Volume 60, Issue 3–4, pp 337–349 | Cite as

Semi-Godunov schemes for general triangular systems of conservation laws

  • Kenneth Hvistendahl Karlsen
  • Siddhartha Mishra
  • Nils Henrik Risebro


General m ×  m triangular systems of conservation laws in one space dimension are considered. These systems arise in applications like multi-phase flows in porous media and are non-strictly hyperbolic. Simple and efficient finite-volume schemes of the Godunov type are devised. These are based on a local decoupling of the system into a series of single conservation laws with discontinuous coefficients and are hence termed semi-Godunov schemes. These schemes are not based on the characteristic structure of the system. Some useful properties of the schemes are derived and several numerical experiments demonstrate their robustness and computational efficiency.


Discontinuous flux Flows in porous media Godunov type schemes Triangular systems 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Gimse T and Risebro NH (1992). Solution of Cauchy problem for a conservation law with discontinuous flux function. SIAM J Math Anal 23(3): 635–648 MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Diehl S (1925). A conservation law with point source and discontinuous flux function modeling continuous sedimentation. SIAM J Appl Math 56(2): 1980–2007 MathSciNetGoogle Scholar
  3. 3.
    Towers JD (2000). Convergence of a difference scheme for conservation laws with a discontinuous flux. SIAM J Numer Anal 38(2): 681–698 MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Towers JD (2001). A difference scheme for conservation laws with a discontinuous flux-the nonconvex case. SIAM J Numer Anal 39(4): 1197–1218 MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Karlsen KH, Risebro NH and Towers JD (2003). Upwind difference approximations for degenerate parabolic convection–diffusion equations with a discontinuous coefficient. IMA J Numer Anal 22(4): 623–664 CrossRefMathSciNetGoogle Scholar
  6. 6.
    Karlsen KH, Risebro NH and Towers JD (2003). L 1 stability for entropy solution of nonlinear degenerate parabolic convection–diffusion equations with discontinuous coefficients. Skr K Nor Vidensk Selsk 3: 49 MathSciNetGoogle Scholar
  7. 7.
    Adimurthi JJ and Veerappa Gowda GD (2004). Godunov-type methods for conservation laws with a flux function discontinuous in space. SIAM J Numer Anal 42(1): 179–208 MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Adimurthi J, Mishra S and Veerappa Gowda GD (2005). Optimal entropy solutions for conservation laws with discontinuous flux. Hyp Diff Eqns 2(4): 1–56 MathSciNetGoogle Scholar
  9. 9.
    Mishra S (2005) Analysis and Numerical approximation of conservation laws with discontinuous coefficients. PhD Thesis, Indian Institute of Science. BangaloreGoogle Scholar
  10. 10.
    Karlsen KH, Mishra S, Risebro NH (2006) Convergence of finite volume schemes for a triangular system of conservation laws. Preprint.Google Scholar
  11. 11.
    Chavent G and Jaffre J (1986). Mathematical models and Finite elements for reservoir simulation. North Holland, Amsterdam MATHGoogle Scholar
  12. 12.
    Bürger R, Karlsen KH, Tory EM and Wendland WL (2002). Model equations and instability regions for the sedimentation of polydisperse suspensions of spheres. ZAMM Z Angew Math Mech 82(10): 699–722 MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Karlsen KH, Lie K-A, Natvig JR, Nordhaug HF and Dahle HK (2001). Operator splitting methods for systems of convection-diffusion equations: nonlinear error mechanisms and correction strategies. J Comput Phys 173(2): 636–663 MATHCrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  • Kenneth Hvistendahl Karlsen
    • 1
  • Siddhartha Mishra
    • 1
  • Nils Henrik Risebro
    • 1
  1. 1.Centre of Mathematics for Applications (CMA)University of OsloBlindernOsloNorway

Personalised recommendations