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Towards a genuinely non-linear accurate Finite Volume scheme with a posteriori stabilization

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Raphaël Loubère (Bordeaux)

The purpose of this work is to build a general framework to reconstruct the underlying fields
within a Finite Volume (FV) scheme solving an hyperbolic system of PDEs (Partial Differential
Equations). In a FV context, the data are piece-wise constants per computational cell and the
physical fields are reconstructed taking into account neighbor cell values. These reconstructions
are further used to evaluate the physical fields which are usually used to feed a Riemann solver.
The physical field reconstructions must obey some physical properties associated to the system of
PDEs (such as the positivity, entropy property, admissibility...) but also some numerically based
properties like an essentially non-oscillatory behavior, computer admissibility representation
(such as NaN, Inf). Moreover the reconstructions should be high accurate for smooth flows and,
at minima, robust/stable on discontinuous solutions and possibly "accurate", ie fews cells of
numerical diffusion.

In this work we introduce a methodology to blend high or low order polynomials and non-linear
hyperbolic tangent reconstructions (THINC functions). A Boundary Variation Diminishing (BVD)
algorithm is employed to select the least dissipative reconstruction in each cell. An a posteriori
MOOD detection procedure is further employed to ensure the physical admissibility and
computer representation in the rare problematic cells (where admissibility is not ensured) by a
local re-computation of the solution with a robust first-order FV scheme.
We illustrate the performance of the proposed scheme via the numerical simulations for some
benchmark tests which involve vacuum or near vacuum states, strong discontinuities, high Mach
flows, etc. The proposed scheme maintains high accuracy on smooth profile, preserves the
positivity and can eliminate the oscillation in the vicinity of discontinuities while maintaining
sharp discontinuity.