[Paper Review] Entropy estimates for a class of schemes for the euler equations

Thierry Gallouët, Raphaèle Herbin+2 more|arXiv (Cornell University)|Jul 5, 2017

Computational Fluid Dynamics and Aerodynamics17 references3 citations

TL;DR

This paper establishes discrete entropy estimates for a class of finite volume schemes solving the Euler equations, using internal energy-based discretization with velocity-based upwinding. It proves that under boundedness in $L^∞$ and BV norms, the entropy inequality is satisfied up to a remainder term that vanishes as space and time steps tend to zero, ensuring convergence to an entropy-satisfying weak solution.

ABSTRACT

In this paper, we derive entropy estimates for a class of schemes for the Euler equations which present the following features: they are based on the internal energy equation (eventually with a positive corrective term at the righ-hand-side so as to ensure consistency) and the possible upwinding is performed with respect to the material velocity only. The implicit-in-time first-order upwind scheme satisfies a local entropy inequality. A generalization of the convection term is then introduced, which allows to limit the scheme diffusion while ensuring a weaker property: the entropy inequality is satisfied up to a remainder term which is shown to tend to zero with the space and time steps, if the discrete solution is controlled in L $\infty$ and BV norms. The explicit upwind variant also satisfies such a weaker property, at the price of an estimate for the velocity which could be derived from the introduction of a new stabilization term in the momentum balance. Still for the explicit scheme, with the above-mentioned generalization of the convection operator, the same result only holds if the ratio of the time to the space step tends to zero.

Motivation & Objective

• To derive discrete entropy inequalities for a class of fully discrete finite volume schemes solving the compressible Euler equations.
• To analyze the consistency and stability of schemes based on the internal energy equation with a corrective term for consistency.
• To establish conditions under which the discrete entropy inequality holds up to a remainder term that vanishes as the mesh and time steps tend to zero.
• To extend entropy stability analysis to both implicit and explicit schemes, particularly in multi-dimensional settings.
• To investigate the role of stabilization terms in the momentum equation to control velocity norms and ensure convergence to an entropy-satisfying limit.

Proposed method

• Formulates the Euler system using the internal energy equation with a positive corrective term to preserve consistency.
• Applies upwinding only with respect to the material velocity in the convection terms, avoiding full upwinding on all variables.
• Introduces a generalized convection operator to reduce numerical diffusion while maintaining entropy-related properties.
• Derives discrete entropy inequalities with a remainder term that depends on the $L^∞$ and BV norms of the discrete solution.
• Uses a time-implicit or time-explicit scheme, with stability ensured via CFL conditions and boundedness assumptions.
• Incorporates stabilization terms in the momentum equation (e.g., $h_{\mathcal{M}}^\alpha \Delta_q u_i$) to control velocity norms and ensure remainder decay.

Experimental results

Research questions

• RQ1 Under what conditions does a fully discrete finite volume scheme for the Euler equations satisfy a discrete entropy inequality up to a remainder term?
• RQ2 How does the remainder term in the entropy inequality behave as the space and time steps tend to zero?
• RQ3 What role does the velocity norm in $L^q(0,T;W^{1,q}_{\mathcal{M}})$ play in ensuring the remainder vanishes?
• RQ4 Can stabilization terms in the momentum equation be designed to ensure the remainder term vanishes without over-diffusing the solution?
• RQ5 How can the proposed scheme be extended to other flux-splitting schemes used in industrial CFD software like CALIF3S?

Key findings

• The implicit first-order upwind scheme satisfies a local discrete entropy inequality without remainder.
• For the generalized convection scheme, the entropy inequality holds up to a remainder term that tends to zero as $h_{\mathcal{M}} \to 0$ and $\delta t \to 0$, provided the discrete solution is bounded in $L^\infty$ and BV norms.
• For the explicit scheme, the remainder term vanishes if the ratio $\delta t / \underline{h}_{\mathcal{M}} \to 0$ or if the velocity norm in $L^q(0,T;W^{1,q}_{\mathcal{M}})$ remains controlled.
• A stabilization term of the form $h_{\mathcal{M}}^\alpha \Delta_q u_i$ in the momentum equation enables control of the velocity norm, with $\alpha < q-1$ required to ensure remainder decay.
• The remainder estimate for the explicit scheme is bounded by $C C_{\mathcal{M}} M^{(2p-1)/p} |\varphi''|_\infty \|\rho\|_{\mathcal{T},t,BV}^{1/p} \|\mathbf{u}\|_{L^q(0,T;W^{1,q}_{\mathcal{M}})} \delta t^{1/p}$, with $1/p + 1/q = 1$.
• The analysis supports convergence to a weak entropy solution under $L^\infty$ and BV stability, provided the remainder vanishes, which is ensured by appropriate stabilization or mesh-time ratio control.