Solution property preserving reconstruction for finite volume scheme: a boundary variation diminishing plus multidimensional optimal order detection framework

The purpose of this work is to build a general framework to reconstruct the underlying fields within a finite volume (FV) scheme solving a hyperbolic system of PDEs (Partial Differential Equations). In an FV context, the data are piecewise 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 states usually used to feed a Riemann solver which computes the associated numerical fluxes. The physical field reconstructions must obey some properties linked to the system of PDEs such as the positivity, but also some numerically based ones, like an essentially nonoscillatory behavior. Moreover, the reconstructions should be highly accurate for smooth flows and robust/stable for discontinuous solutions. To ensure a solution property preserving reconstruction, we introduce a methodology to blend high-/low-order polynomials and hyperbolic tangent reconstructions. A boundary variation diminishing algorithm is employed to select the best reconstruction in each cell. An a posteriori MOOD detection procedure is employed to ensure the positivity by recomputing the rare problematic cells using 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, and also smooth flows. The proposed scheme maintains high accuracy on smooth profile, preserves the positivity and can eliminate the oscillations in the vicinity of discontinuities while maintaining sharper discontinuity with superior solution quality compared to classical highly accurate FV schemes.