Nonlinear Flux Approximation Scheme for Burgers Equation Derived from a Local BVP

  • J. H. M. ten Thije BoonkkampEmail author
  • N. Kumar
  • B. Koren
  • D. A. M. van der Woude
  • A. Linke
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 126)


We present a novel flux approximation scheme for the viscous Burgers equation. The numerical flux is computed from a local two-point boundary value problem for the stationary equation and requires the iterative solution of a nonlinear equation depending on the local boundary values and the viscosity. In the inviscid limit the scheme reduces to the Godunov numerical flux.


  1. 1.
    J.M. Burgers, A Mathematical Model Illustrating the Theory of Turbulence (Academic Press, New York, 1948)CrossRefGoogle Scholar
  2. 2.
    R. Eymard, T. Gallouët, R. Herbin, Finite volume methods, in Handbook of Numerical Analysis, ed. by P.G. Ciarlet, J.L. Lions, vol. VII (North-Holland, Amsterdam, 2000), pp. 713–1020Google Scholar
  3. 3.
    R. Eymard, J. Fuhrmann, K. Gärtner, A finite volume scheme for nonlinear parabolic equations derived from one-dimensional local Dirichlet problems. Numer. Math. 102, 463–495 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    N. Kumar, Flux approximation schemes for flow problems using local boundary value problems, PhD Thesis, Eindhoven University of Technology, 2017Google Scholar
  5. 5.
    N. Kumar, J.H.M. ten Thije Boonkkamp, B. Koren, A. Linke, A nonlinear flux approximation scheme for the viscous Burgers equation, in Finite Volumes for Complex Applications VIII – Hyperbolic, Elliptic and Parabolic Problems, ed. by C. Cances, P. Omnes (Springer, Switzerland, 2017), pp. 457–465CrossRefGoogle Scholar
  6. 6.
    J.H.M. ten Thije Boonkkamp, M.J.H. Anthonissen, The finite volume-complete flux scheme for advection-diffusion-reaction equations. J. Sci. Comput. 46, 47–70 (2011)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • J. H. M. ten Thije Boonkkamp
    • 1
    Email author
  • N. Kumar
    • 1
  • B. Koren
    • 1
  • D. A. M. van der Woude
    • 1
  • A. Linke
    • 2
  1. 1.Department of Mathematics and Computer ScienceEindhoven University of TechnologyEindhovenThe Netherlands
  2. 2.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany

Personalised recommendations