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. K E Y W O R D Sfinite volume, hyperbolic system of PDEs, MOOD, multi-stage-BVD, positivity-preserving, THINC Int J Numer Meth Fluids. 2020;92:603-634.wileyonlinelibrary.com/journal/fld

In "Solution Property Reconstruction for Finite Volume scheme: a BVD+MOOD framework", Int. J. Numer. Methods Fluids, 2020, we have designed a novel solution property preserving reconstruction, so-called multi-stage BVD-MOOD scheme. The scheme is able to maintain a high accuracy in smooth profile, eliminate the oscillations in the vicinity of discontinuity, capture sharply discontinuity and preserve some physical properties like the positivity of density and pressure for the Euler equations of compressible gas dynamics. In this paper, we present an extension of this approach for the compressible Euler equations supplemented with source terms (e.g., gravity, chemical reaction). One of the main challenges when simulating these models is the occurrence of negative density or pressure during the time evolution, which leads to a blow-up of the computation. General compressible Euler equations with different type of source terms are considered as models for physical situations such as detonation waves. Then, we illustrate the performance of the proposed scheme via a numerical test suite including genuinely demanding numerical tests. We observe that the present scheme is able to preserve the physical properties of the numerical solution still ensuring robustness and accuracy when and where appropriate.