Entropy inequalities for fully-discrete E-schemes

  • A. J. KrielEmail author


We consider the numerical solution of one-dimensional scalar conservation laws. In particular, we present some very simple, yet appropriate, discrete entropy fluxes for the class of fully-discrete E-schemes. We show that these provide the required entropy inequalities under sharp CFL conditions.

Mathematics Subject Classification

65M08 65M12 35L65 



  1. 1.
    Harten, A.: High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49(3), 357–393 (1983)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Hopf, E.: On the right weak solution of the cauchy problem for a quasilinear equation of first order. J. Math. Mech. 19(6), 483–487 (1969)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Kružkov, S.N.: First order quasilinear equations in several independent variables. Math. USSR-Sb. 10(2), 217–243 (1970)CrossRefGoogle Scholar
  4. 4.
    Lions, P.L., Perthame, B., Tadmor, E.: A kinetic formulation of multidimensional scalar conservation laws and related equations. J. Am. Math. Soc. 7(1), 169–191 (1994)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Makridakis, C., Perthame, B.: Sharp CFL, discrete kinetic formulation, and entropic schemes for scalar conservation laws. SIAM J. Numer. Anal. 41(3), 1032–1051 (2003)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Osher, S.: Riemann solvers, the entropy condition, and difference approximations. SIAM J. Numer. Anal. 21(2), 217–235 (1984)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Tadmor, E.: Numerical viscosity and the entropy condition for conservative difference schemes. Math. Comput. 43(168), 369–381 (1984)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Applied MathematicsUniversity of the Free StateBloemfonteinSouth Africa

Personalised recommendations