Bad behavior of Godunov mixed methods for strongly anisotropic advection-dispersion equations

  • Authors:

  • Annamaria Mazzia;Gianmarco Manzini;Mario Putti

  • Affiliations:

  • Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Universití degli Studi di Padova, via Trieste 63, 35121 Padova, Italy;Istituto di Matematica Applicata e Tecnologie Informatiche (IMATI), CNR, via Ferrata 1, 27100 Pavia, Italy and Centro di Simulazione Numerica Avanzata (CeSNA), IUSS Pavia, v.le Lungo Ticino Sforza ...;Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Universití degli Studi di Padova, via Trieste 63, 35121 Padova, Italy

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2011

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Abstract

We study the performance of Godunov mixed methods, which combine a mixed-hybrid finite element solver and a Godunov-like shock-capturing solver, for the numerical treatment of the advection-dispersion equation with strong anisotropic tensor coefficients. It turns out that a mesh locking phenomenon may cause ill-conditioning and reduce the accuracy of the numerical approximation especially on coarse meshes. This problem may be partially alleviated by substituting the mixed-hybrid finite element solver used in the discretization of the dispersive (diffusive) term with a linear Galerkin finite element solver, which does not display such a strong ill conditioning. To illustrate the different mechanisms that come into play, we investigate the spectral properties of such numerical discretizations when applied to a strongly anisotropic diffusive term on a small regular mesh. A thorough comparison of the stiffness matrix eigenvalues reveals that the accuracy loss of the Godunov mixed method is a structural feature of the mixed-hybrid method. In fact, the varied response of the two methods is due to the different way the smallest and largest eigenvalues of the dispersion (diffusion) tensor influence the diagonal and off-diagonal terms of the final stiffness matrix. One and two dimensional test cases support our findings.