We propose two novel two-state approximate Riemann solvers for the compressible Euler equations which are provably entropy dissipative and suitable for the simulation of low Mach numbers. What is new, is that one of our two methods in addition is provably kinetic energy stable. Both methods are based on the entropy satisfying and kinetic energy consistent methods of Chandrashekar (2013). The low Mach number compliance is achieved by rescaling some speed of sound terms in the diffusion matrix in the spirit of Li & Gu (2008). In numerical tests we demonstrate the low Mach number compliance and the entropy stability of the proposed fluxes.
Entropy-conserving numerical fluxes are a cornerstone of modern high-order entropy-dissipative discretizations of conservation laws. In addition to entropy conservation, other structural properties mimicking the continuous level such as pressure equilibrium and kinetic energy preservation are important. This note proves that there are no numerical fluxes conserving (one of) Harten's entropies for the compressible Euler equations that also preserve pressure equilibria and have a density flux independent of the pressure. This is in contrast to fluxes based on the physical entropy, where even kin…
This work proposes a high-order hybridised discontinuous Galerkin (HDG) formulation of the Harten-Lax-Van Leer (HLL) Riemann solver for compressible flows. A unified framework is introduced to present Lax-Friedrichs, Roe and HLL Riemann solvers via appropriate definitions of the HDG numerical fluxes. The resulting high-order HDG method with HLL Riemann solver is evaluated through a set of numerical simulations of inviscid compressible flows in different regimes, from subsonic isentropic flows to transonic and supersonic problems with shocks. The accuracy of the proposed method is comparable wi…
We address here the discretization of the momentum convection operator for fluid flow simulations on 2D triangular and quadrangular meshes and 3D polyhedral meshes containing hexahedra, tetrahedra, prisms and pyramids. The finite volume scheme that we use for the full Euler equations is based on a staggered discretization: the density unknowns are associated with a primal mesh, whereas the velocity unknowns are associated with a "fictive" dual mesh. Accordingly, the convection operator of the mass balance equation is derived on the primal mesh, while the the convection operator of the momentum…
This paper introduces a novel approach to compute the numerical fluxes at the cell boundaries for a cell-centered conservative numerical scheme. Explicit gradients used in deriving the reconstruction polynomials are replaced by high-order gradients computed by compact finite differences, referred to as implicit gradients in this paper. A problem-independent shock capturing approach via Boundary Variation Diminishing (BVD) algorithm is used to suppress oscillations for the simulation of flows with shocks and material interfaces. Several numerical test cases are carried out to verify the propose…
This work concerns the numerical approximation of a multicomponent compressible Euler system for a fluid mixture in multiple space dimensions on unstructured meshes with a high-order discontinuous Galerkin spectral element method (DGSEM). We first derive an entropy stable (ES) and robust (i.e., that preserves the positivity of the partial densities and internal energy) three-point finite volume scheme using relaxation-based approximate Riemann solvers from Bouchut [Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources, Birkhauser] a…
This work is in the field of multi-material compressible fluid flows simulation. The proposed scheme is eulerian and related to finite volumes methods, but in a Lagrange-Remap formalism on regular orthogonal meshes. The Lagrangian scheme is staggered and the remap phase is similar to a finite volume advection scheme. The multi-material extension uses classical VOF fluxes for sharp interface reconstruction. The originality of the scheme is in the attempt for a 9 points remap scheme without directional splitting. This strategy should allow to preserve good properties of classical multi-material …
This paper describes a class of scheme named "residual distribution schemes" or "fluctuation splitting schemes". They are a generalization of Roe's numerical flux in fluctuation form. The so-called multidimensional fluctuation schemes have historically first been developed for steady homogeneous hyperbolic systems. Their application to unsteady problems and conservation laws has been really understood only relatively recently. This understanding has allowed to make of the residual distribution framework a powerful playground to develop numerical discretizations embedding some prescribed constr…
We present a numerical approximation of the solutions of the Euler equations with a gravitational source term. On the basis of a Suliciu type relaxation model with two relaxation speeds, we construct an approximate Riemann solver, which is used in a first order Godunov-type finite volume scheme. This scheme can preserve both stationary solutions and the low Mach limit to the corresponding incompressible equations. In addition, we prove that our scheme preserves the positivity of density and internal energy, that it is entropy satisfying and also guarantees not to give rise to numerical checker…
This paper introduces a novel approach to compute the numerical fluxes at the cell boundaries in the finite volume approach. Explicit gradients used in deriving the reconstruction polynomials are replaced by high-order gradients computed by compact finite differences, referred to as implicit gradients in this paper. The new finite volume scheme has superior dispersion and dissipation properties in comparison to the compact reconstruction approach. These implicit gradients are re-used in viscous flux computation and post-processing, which further improves efficiency. A problem-independent shock…
Well-Balanced Schemes for the Euler Equations with Gravitation: Conservative Formulation Using Global Fluxes
North Carolina State University（北卡罗来纳州立大学）；University of Maryland（马里兰大学）；Temple University（天普大学）；Southern University of Science and Technology of China（南方科技大学）；Tulane University（杜兰大学）
Positivity-Preserving and Entropy-Stable Oscillation-Eliminating DGSEM for the Compressible Euler Equations on Curvilinear Meshes with Adaptive Mesh Refinement
In this work, Entropy-Stable (ES) schemes are formulated for the multicomponent compressible Euler equations. Entropy-conservative (EC) and ES fluxes are derived. Particular attention is paid to the limit case of zero partial densities where the structure required by ES schemes breaks down (the entropy variables are no longer defined). It is shown that while an EC flux is well-defined in this limit, a well-defined upwind ES flux requires appropriately averaged partial densities in the dissipation matrix. A similar result holds for the high-order TecNO reconstruction. However, this does not pre…
Low Mach number limit of a pressure correction MAC scheme for compressible barotropic flows
Aix-Marseille Université, CNRS, École Centrale de Marseille（艾克斯-马赛大学，法国国家科学研究中心，马赛中央理工学院）；Institut de Radioprotection et de Sûreté Nucléaire (IRSN)（核辐射防护与安全研究所）；Université de Lyon, CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan（里昂大学，法国国家科学研究中心UMR 5208，里昂第一大学，卡米尔·若尔当研究所）
Entropy-Stable and Well-Balanced Discontinuous Galerkin Methods for the Compressible Euler Equations in Vector-Invariant Form
Johannes Gutenberg University Mainz（约翰内斯·古腾堡美因茨大学）；Australian National University（澳大利亚国立大学）；Leibniz Institute for Tropospheric Research (TROPOS)（莱布尼茨对流层研究所）
A novel notion for constructing a well-balanced scheme - a gradient-robust scheme - is introduced and a showcase application for a steady compressible, isothermal Stokes equations is presented. Gradient-robustness means that arbitrary gradient fields in the momentum balance are well-balanced by the discrete pressure gradient - if there is enough mass in the system to compensate the force. The scheme is asymptotic-preserving in the sense that it degenerates for low Mach numbers to a recent inf-sup stable and pressure-robust discretization for the incompressible Stokes equations. The convergence…
A fully conservative sharp-interface method is developed for multiphase flows with phase change. The coupling between two phases is implemented via introducing the interfacial fluxes, which are obtained by solving a general Riemann problem with phase change. A novel four-wave model is proposed to obtain an approximate Riemann solution, which simplifies the eight-dimensional roo-finding procedure in the exact solver to a sole iteration of the mass flux. Unlike in the previous research, the jump conditions of all waves are imposed strictly in the present approximate Riemann solver so that conser…
Structure-Preserving High-Order Methods for the Compressible Euler Equations in Potential Temperature Formulation for Atmospheric Flows
Institute of Mathematics, Johannes Gutenberg University Mainz（美因茨约翰内斯·古腾堡大学数学研究所）；Leibniz Institute for Tropospheric Research (TROPOS)（莱布尼茨对流层研究所）；Institute for Atmospheric Physics, Johannes Gutenberg University Mainz（美因茨约翰内斯·古腾堡大学大气物理研究所）
Low Mach number limit of some staggered schemes for compressible barotropic flows
Aix-Marseille Université, CNRS, École Centrale de Marseille（艾克斯-马赛大学、法国国家科学研究中心、马赛中央理工学院）；Institut de Radioprotection et de Sûreté Nucléaire (IRSN)（核辐射防护与核安全研究所）；Université de Lyon, CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan（里昂大学、法国国家科学研究中心、里昂第一大学、卡米尔·若尔当研究所）
Fully compressible magnetohydrodynamic (MHD) simulations are a fundamental tool for investigating the role of dynamo amplification in the generation of magnetic fields in deep convective layers of stars. The flows that arise in such environments are characterized by low (sonic) Mach numbers (M_son < 0.01 ). In these regimes, conventional MHD codes typically show excessive dissipation and tend to be inefficient as the Courant-Friedrichs-Lewy (CFL) constraint on the time step becomes too strict. In this work we present a new method for efficiently simulating MHD flows at low Mach numbers in a sp…
