An Adaptive E-Scheme for Conservation Laws

  • Ebise A. Abdi
  • Christian V. Hansen
  • H. Joachim SchrollEmail author
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 126)


An adaptive E-scheme for possibly degenerate, viscous conservation laws is presented. The scheme makes use of both given and numerical diffusion to establish the E-property. In the degenerate case it reduces to local Lax–Friedrichs. Both explicit and time-implicit E-schemes are monotone and TVD. Numerical experiments demonstrate the robustness and improved accuracy of the adaptive scheme.



The authors acknowledge funding by the Ethiopian Department of Education, and the Norwegian Agency for Development Cooperation.


  1. 1.
    E.A. Abdi, C.V. Hansen, H.J. Schroll, An adaptive viscosity E-scheme for balance laws. Preprint SDU (2017)Google Scholar
  2. 2.
    M.G. Crandall, A. Majda, Monotone difference approximations for scalar conservation laws. Math. Comput. 34(149), 1–21 (1980)MathSciNetCrossRefGoogle Scholar
  3. 3.
    H.O. Kreiss, J. Lorenz, Initial–Boundary Value Problems and the Navier–Stokes Equations (Academic, Bosten, 1989)zbMATHGoogle Scholar
  4. 4.
    R.J. LeVeque, Numerical Methods for Conservation Laws (Birkhäuser, Basel, 1992)Google Scholar
  5. 5.
    S. Osher, Riemann solvers, the entropy condition, and difference approximations. SIAM J. Numer. Anal. 21, 217–235 (1984)MathSciNetCrossRefGoogle Scholar
  6. 6.
    V.V. Rusanov, Calculation of interaction of nonsteady shock waves with obstacles. J. Comput. Math. Phys. USSR 1, 267–279 (1961)Google Scholar
  7. 7.
    E. Tadmor, Numerical viscosity and the entropy condition for conservative difference schemes. Math. Comput. 43, 369–381 (1984)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Ebise A. Abdi
    • 1
  • Christian V. Hansen
    • 2
  • H. Joachim Schroll
    • 2
    Email author
  1. 1.School of Mathematical and Statistical SciencesHawassa UniversityHawassaEthiopia
  2. 2.Department of Mathematics and Computer ScienceUniversity of Southern DenmarkOdense MDenmark

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