Selection criteria for the compressible Euler equations: numerical study

2026/08/03 by Megala Anandan, Bahulayan Chalil Aravind, Mária Lukáčová-Medvid'ová +4
Physics and Astronomy · Computer Science · Mathematics · #math-ph #cs.NA #math.MP #math.NA #msc:65M08 #msc:65M60 #msc:76M10 #msc:76M12 #msc:76N10 #msc:35L65

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arxiv created 2026/08/03 · arxiv updated 2026/08/05

Abstract

We numerically investigate selection criteria for dissipative weak solutions of the compressible Euler equations. Using the Kelvin-Helmholtz problem as a benchmark, we compare four standard numerical schemes - the viscous finite volume (VFV) method, the discrete velocity Boltzmann finite volume (DVBFV) method, the Runge-Kutta discontinuous Galerkin (RKDG) method, and the discontinuous Galerkin spectral element method (DGSEM), and demonstrate that each may converge to a different dissipative weak solution. We evaluate these solutions with respect to several selection criteria based on entropy production, total energy, and energy defect, and examine the sensitivity of selection functionals to the numerical diffusion inherent in each scheme.

Citations

• Well-posedness of the Euler system of gas dynamics
• Oscillatory approximations and maximum entropy principle for the Euler system of gas dynamics
• High-Order Flux Splitting Schemes for the Euler Equations of Gas Dynamics
• Maximal turbulence as a selection criterion for measure-valued solutions
• Maximal dissipation and well-posedness of the Euler system of gas dynamics
• What is a limit of structure-preserving numerical methods for compressible flows?
• Consistency and convergence of flux-corrected finite element methods for nonlinear hyperbolic problems
• On the convergence of residual distribution schemes for the compressible Euler equations via dissipative weak solutions
• Convergence of Discontinuous Galerkin Schemes for the Euler Equations via Dissipative Weak Solutions
• Convergence of first-order Finite Volume Method based on Exact Riemann Solver for the Complete Compressible Euler Equations
• Computing oscillatory solutions of the Euler system via K-convergence
• Generalized solutions to models of inviscid fluids
• On convergence of approximate solutions to the compressible Euler system
• K-convergence as a new tool in numerical analysis
• Non-uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial data
• A finite volume scheme for the Euler system inspired by the two velocities approach
• Convergence of finite volume schemes for the Euler equations via\n dissipative measure--valued solutions
• Maximal dissipation principle for the complete Euler system
• Measure-valued solutions to the complete Euler system revisited
• On the energy dissipation rate of solutions to the compressible isentropic Euler system
• Global ill-posedness of the isentropic system of gas dynamics
• A counterexample to well-posedness of entropy solutions to the compressible Euler system
• On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes
• On admissibility criteria for weak solutions of the Euler equations
• A New Class of Optimal High-Order Strong-Stability-Preserving Time Discretization Methods
• The Runge–Kutta Discontinuous Galerkin Method for Conservation Laws V
• On positivity-preserving high order discontinuous Galerkin schemes for compressible Navier–Stokes equations
• Strong Stability-Preserving High-Order Time Discretization Methods