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A skew-symmetric energy and entropy stable formulation of the compressible Euler equations

Jan Nordström

TL;DR

The paper develops a nonlinear stability framework for hyperbolic problems by reformulating the compressible Euler equations in a skew-symmetric form that yields energy and entropy bounds. It introduces a new energy norm with variables $\boldsymbol{\Phi}$ (denoted here as $\boldsymbol{\nu}$ in the Euler context) and constructs explicit matrices $A_1,A_2,B_1,B_2$ alongside a norm matrix $P$ so that the continuous system satisfies an energy rate identity; the boundary terms govern the energy exchange with the domain. The authors then show how to discretize the skew-symmetric formulation with SBP operators to obtain a semi-discrete scheme that conserves or dissipates energy and entropy in accordance with the boundary treatment, providing a robust approach for nonlinear stability of the Euler equations. They also analyze nonlinear boundary conditions, discuss the dependence on free parameters, and outline open questions and future work, including extensions to viscous terms and related systems like the shallow water equations. Overall, the work offers a unified continuous-and-discrete pathway to energy- and entropy-stable schemes for nonlinear hyperbolic flow problems.

Abstract

We show that a specific skew-symmetric form of nonlinear hyperbolic problems leads to energy and entropy bounds. Next, we exemplify by considering the compressible Euler equations in primitive variables, transform them to skew-symmetric form and show how to obtain energy and entropy estimates. Finally we show that the skew-symmetric formulation lead to energy and entropy stable discrete approximations if the scheme is formulated on summation-by-parts form.

A skew-symmetric energy and entropy stable formulation of the compressible Euler equations

TL;DR

The paper develops a nonlinear stability framework for hyperbolic problems by reformulating the compressible Euler equations in a skew-symmetric form that yields energy and entropy bounds. It introduces a new energy norm with variables (denoted here as in the Euler context) and constructs explicit matrices alongside a norm matrix so that the continuous system satisfies an energy rate identity; the boundary terms govern the energy exchange with the domain. The authors then show how to discretize the skew-symmetric formulation with SBP operators to obtain a semi-discrete scheme that conserves or dissipates energy and entropy in accordance with the boundary treatment, providing a robust approach for nonlinear stability of the Euler equations. They also analyze nonlinear boundary conditions, discuss the dependence on free parameters, and outline open questions and future work, including extensions to viscous terms and related systems like the shallow water equations. Overall, the work offers a unified continuous-and-discrete pathway to energy- and entropy-stable schemes for nonlinear hyperbolic flow problems.

Abstract

We show that a specific skew-symmetric form of nonlinear hyperbolic problems leads to energy and entropy bounds. Next, we exemplify by considering the compressible Euler equations in primitive variables, transform them to skew-symmetric form and show how to obtain energy and entropy estimates. Finally we show that the skew-symmetric formulation lead to energy and entropy stable discrete approximations if the scheme is formulated on summation-by-parts form.
Paper Structure (14 sections, 2 theorems, 42 equations)

This paper contains 14 sections, 2 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. The main theory
  3. Energy analysis
  4. Entropy analysis
  5. The new skew-symmetric form of the compressible Euler equations
  6. Task 1: Find an appropriate norm and new dependent variables
  7. Task 2: Rewriting the compressible Euler equations in new dependent variables
  8. Task 3: Transforming the new set equations into a skew-symmetric form
  9. Task 4: Applying the energy method such that only boundary terms remain
  10. Task 5: The choice of nonlinear boundary conditions
  11. The final form of the governing equations
  12. Some open questions
  13. A stable energy and entropy conserving numerical approximation
  14. Summary, conclusions and outlook

Key Result

Proposition 2.1

The IBVP (eq:nonlin) for linear ($V \neq U$) and nonlinear ($V=U$) is energy conserving if holds. It is energy bounded if it is energy conserving and the boundary conditions $L_p U=0$ are such that

Theorems & Definitions (10)

  • Definition 2.1
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Remark 3.1
  • Remark 4.1