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Structure-preserving schemes conserving entropy and kinetic energy

Kunal Bahuguna, Ramesh Kolluru, S. V. Raghurama Rao

TL;DR

The paper addresses the challenge of simulating compressible Euler flows by developing structure-preserving schemes that conserve entropy and kinetic energy in a semi-discrete sense. It introduces entropy-conservative fluxes (EC1, EC2) and a kinetic-energy-preserving flux (ECKEP), and then couples these with entropy-stable diffusion via a Rankine–Hugoniot framework and a hybrid entropy-stable scheme that uses an entropy-distance shock sensor. The key contributions include a diffusion-based approach that ties entropy conservation to the energy equation, a second-order, non-logarithmic EC/KE-preserving flux family, and a practical hybrid scheme that maintains accuracy in smooth regions while stabilizing shocks. The methods are validated on a range of 1D and 2D Riemann and flow-over-aero benchmarks, demonstrating robust shock-capturing, exact contact discontinuity preservation, and avoidance of common instabilities such as carbuncle and odd-even decoupling, with favorable computational efficiency.

Abstract

This paper presents a novel structure-preserving scheme for Euler equations, focusing on the numerical conservation of entropy and kinetic energy. Explicit flux functions engineered to conserve entropy are introduced within the finite-volume framework. Further, discrete kinetic energy conservation too is introduced. A systematic inquiry is presented, commencing with an overview of numerical entropy conservation and formulation of entropy-conserving and kinetic energy-preserving fluxes, followed by the study of their properties and efficacy. A novelty introduced is to associate numerical entropy conservation to the discretization of the energy conservation equation. Furthermore, an entropy-stable shock-capturing diffusion method and a hybrid approach utilizing the entropy distance to manage smooth regions effectively are also introduced. The addition of artificial viscosity in appropriate regions ensures entropy generation sufficient to prevent numerical instabilities. Various test cases, showcasing the efficacy and stability of the proposed methodology, are presented.

Structure-preserving schemes conserving entropy and kinetic energy

TL;DR

The paper addresses the challenge of simulating compressible Euler flows by developing structure-preserving schemes that conserve entropy and kinetic energy in a semi-discrete sense. It introduces entropy-conservative fluxes (EC1, EC2) and a kinetic-energy-preserving flux (ECKEP), and then couples these with entropy-stable diffusion via a Rankine–Hugoniot framework and a hybrid entropy-stable scheme that uses an entropy-distance shock sensor. The key contributions include a diffusion-based approach that ties entropy conservation to the energy equation, a second-order, non-logarithmic EC/KE-preserving flux family, and a practical hybrid scheme that maintains accuracy in smooth regions while stabilizing shocks. The methods are validated on a range of 1D and 2D Riemann and flow-over-aero benchmarks, demonstrating robust shock-capturing, exact contact discontinuity preservation, and avoidance of common instabilities such as carbuncle and odd-even decoupling, with favorable computational efficiency.

Abstract

This paper presents a novel structure-preserving scheme for Euler equations, focusing on the numerical conservation of entropy and kinetic energy. Explicit flux functions engineered to conserve entropy are introduced within the finite-volume framework. Further, discrete kinetic energy conservation too is introduced. A systematic inquiry is presented, commencing with an overview of numerical entropy conservation and formulation of entropy-conserving and kinetic energy-preserving fluxes, followed by the study of their properties and efficacy. A novelty introduced is to associate numerical entropy conservation to the discretization of the energy conservation equation. Furthermore, an entropy-stable shock-capturing diffusion method and a hybrid approach utilizing the entropy distance to manage smooth regions effectively are also introduced. The addition of artificial viscosity in appropriate regions ensures entropy generation sufficient to prevent numerical instabilities. Various test cases, showcasing the efficacy and stability of the proposed methodology, are presented.
Paper Structure (34 sections, 84 equations, 28 figures, 5 tables)

This paper contains 34 sections, 84 equations, 28 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Structures within Euler Equations: Entropy and Kinetic Energy
  3. Entropy inequality, entropy variables, symmetric form and entropy potential functions
  4. Structures in the total energy equation: entropy conservation and kinetic energy preservation
  5. Numerical Entropy Conservation
  6. Numerical Kinetic Energy Preservation
  7. Entropy Conservative and Kinetic Energy Preserving Flux
  8. Entropy Conservative Numerical Flux
  9. Entropy Conservative and Kinetic Energy Preserving Numerical Flux
  10. Numerical investigation of the entropy conservative and kinetic energy preserving fluxes
  11. Exact Capturing of Steady Contact Discontinuity
  12. Experimental order of convergence (EOC)
  13. Conservation properties of the new flux functions
  14. Taylor-Green Vortex
  15. Isentropic vortex convection
  16. ...and 19 more sections

Figures (28)

  • Figure 1: Density and velocity plots for a stationary contact discontinuity at T=2 s by the three entropy conservative fluxes
  • Figure 2: Log-log plot of errors with grid size for EC1, EC2 and ECKEP fluxes respectively
  • Figure 3: Total Entropy for Taylor-Greens vortex
  • Figure 4: Total Kinetic Energy for Taylor-Greens vortex
  • Figure 5: Total entropy and total kinetic energy v/s Time for Isentropic vortex test case for different entropy conservative fluxes
  • ...and 23 more figures