Solution property preserving reconstruction BVD+MOOD scheme for compressible euler equations with source terms and detonations

Abstract

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.

Symmetry-preserving enforcement of low-dissipation method based on boundary variation diminishing principle

“…THINC scheme was developed for capturing moving interfaces or free boundaries in multiphase flows [29,30,47,48,31]. Thanks to its monotonicity and capability to mimic the step-like profile, the THINC reconstruction is also well suited for computing compressible flows including discontinuous solutions with appealing stability and high resolution [22,23,25,24,26,27]. In the THINC scheme, the physical quantity q is reconstructed by a hyperbolic tangent function as…”

“…Based on the observation that the dissipation term in the approximate Riemann solvers can be effectively reduced by minimizing the difference between the reconstructed left-and right-side values at cell boundaries (referred to as BV (boundary variation)), a new class of reconstruction schemes so-called BVD (Boundary Variation Diminishing) schemes have been proposed in [18,19,20,21,22,23,24,25,26,27,28]. The first BVD scheme [18], so-called WENO-THINC-BVD scheme, switches the reconstruction function to WENO scheme for smooth solutions and THINC (Tangent of Hyperbola for INterface Capturing) for discontinuous solutions.…”

“…As a result, the numerical solution may lose spatial symmetry even started from a symmetrical setup. It is commonly observed that highresolution simulations with fine meshes usually lead to undesirable asymmetrical numerical results [34,35,36,37,13,38,39,40,33,41,42,26] in application problems that possess physical symmetry in solution structures, such as the Rayleigh-Taylor instability (RTI) and the symmetric implosion benchmark problems. The symmetry-breaking results might not be visible in simulations with significant numerical dissipation, and have not been seen as a serious problem in the past.…”

Positivity‐preserving scheme for the flux reconstruction framework with a novel implementation

“…Remarkable works such as References 5‐7 report great improvement of robustness owing to its application. The positivity‐preserving property may be realized through a posteriori correction or reconstruction, 8 and extended to more problems 9‐12 . In References 13‐15, a popular positivity‐preserving scheme is developed for solving different equations on various types of meshes.…”