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

RaphaŽl 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.

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