About Non Linear Stabilization for Scalar Hyperbolic Problems

  • Rémi AbgrallEmail author
Part of the Fields Institute Communications book series (FIC, volume 79)


This paper deals with the numerical approximation of linear and non linear hyperbolic problems. We are mostly interested in the development of parameter free methods that satisfy a local maximum principle. We focus on the scalar case, but extensions to systems are relatively straightforward when these techniques are combined with the ideas contained in Abgrall (J. Comput. Phys., 214(2):773–808, 2006). In a first step, we precise the context, give conditions that guaranty that, under standard stability assumptions, the scheme will converge to weak solutions. In a second step, we provide conditions that guaranty an arbitrary order of accuracy. Then we provide several examples of such schemes and discuss in some details two versions. Numerical results support correctly our initial requirements: the schemes are accurate and satisfy a local maximum principle, even in the case of non smooth solutions.



This work has been partially funded by the SNF grant # 200021_153604 of the Swiss National Foundation. Stimulating discussions with Dr. A. Burbeau (CEA, France), are also warmly acknowledged.


  1. 1.
    R. Abgrall. Essentially non-oscillatory residual distribution schemes for hyperbolic problems. J. Comput. Phys., 214(2):773–808, 2006.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    R. Abgrall. A residual method using discontinuous elements for the computation of possibly non smooth flows. Adv. Appl. Math. Mech, 2010.Google Scholar
  3. 3.
    R. Abgrall and D. de Santis. High-order preserving residual distribution schemes for advection-diffusion scalar problems on arbitrary grids. SIAM I. Sci. Comput., 36(3):A955–A983, 2014. also
  4. 4.
    R. Abgrall and D. de Santis. Linear and non-linear high order accurate residual distribution schemes for the discretization of the steady compressible navier-stokes equations. J. Comput. Phys., 283:326–359, 2015.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    R. Abgrall, A. Larat, and M. Ricchiuto. Construction of very high order residual distribution schemes for steady inviscid flow problems on hybrid unstructured meshes. J. Comput. Phys., 230(11):4103–4136, 2011.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    R. Abgrall and P. L. Roe. High-order fluctuation schemes on triangular meshes. J. Sci. Comput., 19(1–3):3–36, 2003.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    R. Abgrall and C.W. Shu. Development of residual distribution schemes for discontinuous galerkin methods. Commun. Comput. Phys., 5:376–390, 2009.MathSciNetzbMATHGoogle Scholar
  8. 8.
    R. Abgrall, S. Tokereva, and J. Nordström. Entropy stable fem methods. in preparation.Google Scholar
  9. 9.
    R. Abgrall and J. Trefilick. An example of high order residual distribution scheme using non-lagrange elements. J. Sci. Comput., 45(1–3):3–25, 2010.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    E. Burman and P. Hansbo. Edge stabilisation for galerkin approximations of convection-diffusion-reaction problems. Comput. Methods Appl. Mech. Engrg., 193:1437–1453, 2004.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    B. Cockburn, S. Hou, and C.-W. Shu. TVB Runge-Kutta local projection discontinuous finite element method for conservation laws IV: the multidimensional case. Math. Comp., 54:545–581, 1990.MathSciNetzbMATHGoogle Scholar
  12. 12.
    B. Cockburn and C.-W. Shu. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: General framework. Math. Comp., 52:411–435, 1989.MathSciNetzbMATHGoogle Scholar
  13. 13.
    A. Ern and J.L. Guermond. Finite element quasi-interpolation and best approximation. arXiv:1505.06931, May 2015.Google Scholar
  14. 14.
    C. Geuzaine and J.-F. Remacle. A three-dimensional finite element mesh generator with built-in pre- and post-processing facilities.
  15. 15.
    A. Harten. On a class of high resolution total-variational-stable finite-difference schemes (with appendix by Peter D. Lax). SIAM J. Numer. Anal., 21:1–23, 1984.Google Scholar
  16. 16.
    T.J.R. Hughes, L.P. Franca, and M. Mallet. A new finite element formulation for CFD: I. symmetric forms of the compressible Euler and Navier-Stokes equations and the second law of thermodynamics. Comp. Meth. Appl. Mech. Engrg., 54:223–234, 1986.Google Scholar
  17. 17.
    C. Johnson, U. Nävert, and J. Pitkäranta. Finite element methods for linear hyperbolic problems. Computer methods in applied mechanics and engineering, 45:285–312, 1985.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    N. Kroll, H. Bieler, H. Deconinck, V. Couaillier, H. van der Ven, and K. Sorensen, editors. ADIGMA- A European Initiative on the Development of Adaptive Higher-Order Variational Methods for Aerospace Applications. Notes on Numerical Fluid Mechanics and Multidisciplinary Design. Springer, 2010. Results of a Collaborative Research Project Funded by the European Union, 2006–2009.Google Scholar
  19. 19.
    D. Kröner, M. Rokyta, and M. Wierse. A Lax-Wendroff type theorem for upwind finite volume schemes in 2-d. East-West J. Numer. math., 4(4):279–292, 1996.MathSciNetzbMATHGoogle Scholar
  20. 20.
    R.-H. Ni. A multiple grid scheme for solving the Euler equations. In 5th Computational Fluid Dynamics Conference, pages 257–264, 1981.Google Scholar
  21. 21.
    P.L. Roe. Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys., 43:357–372, 1981.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    P.L. Roe. Characteristic-based schemes for the Euler equations. Annu. Rev. Fluid Mech. 18, 337–365 (1986)., 1986.zbMATHGoogle Scholar
  23. 23.
    R. Struijs, H. Deconinck, and P.L. Roe. Fluctuation splitting schemes for the 2D Euler equations. VKI-LS 1991-01, 1991. Computational Fluid Dynamics.Google Scholar

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Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of ZürichZurichSwitzerland

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