Adaptive Flux Corrected Transport FEM for hyperbolic equations stemming from conservation laws

  • Devos, Thibaut (Mines Paris - CEMEF)
  • Hachem, Elie (Mines Paris - CEMEF)
  • Larcher, Aurélien (Mines Paris - CEMEF)

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Shocks and discontinuities present in compressible flows at high Mach number are in general a source of numerical instability due to the strong hyperlocity. Also, the positivity of the density, the mass conservation and the invariant-domain preservation are properties that must be preserved in order to handle correctly these phenomena. In this work, we derived an adaptive finite element method for conservative equations. The method is based on two solutions. First, a low order solution respecting the maximum principle is computed thanks to the addition of a strong diffusive term. This solution is over diffusive, however, it presents the advantage to prevent the apparition of any under/over-shoots. On the other hand, a high-order solution can be computed which is not maximum preserving, but can be made so after some limiting and that still preserves its high-order property. The limiting is based on the Boris-Book-Zalesak flux corrected technique. Finally, the method is combined with an a posteriori error estimator for dynamic anisotropic mesh adaptation. It involves building a mesh based on a metric map. It provides both the size and the stretching of elements in a very condensed information data. Consequently, due to the presence of high gradients in the solved variable, it provides highly stretched elements at the shock interfaces and at the boundary layers, and thus yields an accurate modeling framework for conservative equations. Both implicit and explicit time schemes are implemented. Forward Euler and Runge-Kutta schemes are compared in explicit framework.