Positivity-preserving hybrid DG/FV method for compressible Euler equations with stiff source terms

TL;DR

This paper introduces a hybrid DG/FV numerical method that preserves positivity in simulations of compressible flows with stiff source terms, effectively preventing unphysical negative densities or pressures.

Contribution

The proposed hybrid DG/FV method combines a priori and a posteriori techniques, including relaxed BVD and MOOD paradigms, to ensure positivity and reduce numerical dissipation in complex fluid simulations.

Findings

Successfully prevents negative density and pressure in simulations.

Reduces oscillations and numerical dissipation compared to traditional methods.

Maintains small-scale resolution and sharp discontinuity capturing capabilities.

Abstract

In numerical simulations of complex fluid dynamical problems, unphysical negative density or pressure may appear, causing blow-up of the computation. With the aim of obtaining positivity-preserving solutions with multi-scale resolution for solving strongly compressible flows, including hypersonic flows and stiff detonation waves, we present a positivity-preserving hybrid discontinuous Galerkin/finite volume (DG/FV) method. The approach is based on a priori and a posteriori computational methodology. The a priori computation utilizes relaxed boundary variation diminishing (BVD) algorithm to find troubled cells where the DG operators are replaced by the FV operators. The FV operators then deploy a hyperbolic tangent function in the reconstruction procedure to prevent unphysical values appearing in the flux evaluation. The a posteriori computation detects unphysical negative values in a simplified version of multidimensional optimal order detection (MOOD) paradigm and in the worst case applies the first-order FV Godunov scheme to guarantee the positivity of the solution. The a priori computation produces fewer oscillatory solutions, so that the a posteriori computation triggers less first-order evaluation. A technique of utilizing the tangent of hyperbola for interface capturing (THINC) with anti-diffusion effect, which is also referred to as the technique of adaptive reconstruction in this work, is suggested to reduce the numerical dissipation of the shock-capturing scheme. The current approaches retain the capability of the DG scheme to resolve small scales and the capability of the FV scheme to capture sharp discontinuities at the subgrid level.