Viscous regularization of the Euler equations and entropy principles

Jean-Luc Guermond; Bojan Popov

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Viscous regularization of the Euler equations and entropy principles

Jean-Luc Guermond, Bojan Popov

arXiv:1212.5566v1Dec 21, 2012math-phmath.MP

TL;DR

Preserving density and internal-energy positivity, a minimum specific-entropy principle, and broad entropy inequalities is difficult for Euler approximations. The paper investigates a general viscous regularization class and identifies a unique choice meeting the entropy criteria, while connecting it to parabolic and Navier–Stokes formulations.

Problem
Few numerical schemes on arbitrary meshes in two or more dimensions are known to preserve density and internal-energy positivity alongside a minimum specific-entropy principle.
Source: §1 · p. 1
Method
The paper analyzes a general family of Euler viscous regularizations for the minimum entropy principle and compatibility with generalized entropies.
Source: §1 · p. 2, §5.3 · p. 15
Results
The choice a = d is the unique regularization compatible with all generalized entropies and the minimum entropy principle, and the formulation bridges parabolic and Navier–Stokes regularizations.
Source: §4.1 · p. 11, §5 · p. 13, §5.3 · p. 15
Takeaways & Limitations
Distinguishing mass and volume velocities rewrites the regularization in a form similar to Navier–Stokes with rotation-invariant viscous fluxes.
Source: §1 · p. 2
The analysis assumes the stated convexity condition and positive temperature, without requiring positive pressure.
Source: §2.1 · p. 3

Abstract

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This paper investigates a general class of viscous regularizations of the compressible Euler equations. A unique regularization is identified that is compatible with all the generalized entropies a la Harten and satisfies the minimum entropy principle. A connection with a recently proposed phenomenological model by H. Brenner is made.

1. Introduction.

The introduction frames the challenge of preserving positivity and a minimum specific-entropy principle in Euler approximations, especially on arbitrary multidimensional meshes. It asks whether a Galilean-invariant regularization can meet those properties and broad entropy requirements, and previews a unique formulation and its connection to Brenner’s model.

1. Introduction.

Few schemes on arbitrary meshes in two or more dimensions are known to preserve density and internal-energy positivity together with a minimum specific-entropy principle.The introduction notes that arguments for Godunov and Lax variants do not readily extend to central high-order or Galerkin schemes.Source: §1 · p. 1
The paper asks for a Galilean-invariant Euler regularization that ensures density and internal-energy positivity, a minimum specific-entropy principle, and compatibility with many entropy inequalities.The standard parabolic regularization acts on conserved variables, while Navier–Stokes lacks continuity-equation regularization and can violate the minimum principle when thermal diffusivity is nonzero.Source: §1 · pp. 1–2
Only one regularization satisfies the minimum entropy principle and is compatible with all generalized entropies of Harten et al..The paper identifies this as its key result.Source: §1 · p. 2
The formulation is compared with Brenner’s heuristic model and, by distinguishing mass and volume velocities, can be written with rotation-invariant viscous fluxes in a form similar to Navier–Stokes.The discussion presents parabolic and Navier–Stokes regularizations as two views of the same formulation.Source: §1 · p. 2

2. Standard regularizations.

The section reviews standard regularizations and their trade-offs, then states the thermodynamic setting for the proposed Euler regularization. Monolithic parabolic viscosity and Navier–Stokes viscosity each have limitations for the desired numerical properties.

2. Standard regularizations.

The section reviews regularization techniques for the compressible Euler equations and discusses their respective advantages and drawbacks.The system is stated in conservative variables: density, momentum, and total energy.Source: §2 · p. 2, §2.1 · p. 2

2.1. Statement of the problem.

The proposed Euler regularization is intended to be compatible with thermodynamics and serve as a starting point for numerical approximation; the setup assumes convexity and positive temperature, but not positive pressure.The stated sufficient thermodynamic assumptions are convexity condition (2.8) and positive temperature.Source: §2.1 · p. 3

2.2. Monolithic parabolic regularization.

The monolithic parabolic regularization applies εΔ indiscriminately to conserved quantities, and its numerical viscosity can be over-dissipative when the CFL number appears in the denominator.The parabolic approach also appears to violate Galilean and rotation invariance because it dissipates density, momentum, and total energy.Source: §2.2 · pp. 3–4

2.3. Navier-Stokes regularization.

The Navier–Stokes regularization uses viscous and thermal fluxes, but it does not regularize the continuity equation.The authors identify the missing density regularization as an obstruction for numerical purposes.Source: §2.3 · p. 4
With nonzero thermal diffusivity, Navier–Stokes can violate the minimum entropy principle and fail to admit generalized entropy inequalities; with zero diffusivity, contact waves lack the desired viscous regularization.The authors conclude that this regularization is not appropriate when density positivity, the minimum principle, and reasonable contact-discontinuity approximation are desired.Source: §2.3 · pp. 4–5

3. General regularization.

The section frames viscous regularization as a flux-design problem: choose mass, momentum, and energy fluxes to target positive density, a minimum specific-entropy principle, and compatibility with a large class of entropies.

3. General regularization.

The general framework treats fluxes f, g, and h as design variables for positive density, a minimum specific-entropy principle, and compatibility with a large class of entropies.The section assumes smooth solutions; it states that the internal energy will be positive as a consequence of positive density and the minimum entropy principle.Source: §3 · pp. 5–6

3.1. Positivity of the density.

For mass conservation, the authors consider f = a(ρ, e)∇ρ with smooth positive a as a candidate regularization for preserving density positivity.The passage presents this choice as a conjectured appropriate form, not as a proven outcome in the supplied text.Source: §3.1 · p. 6

3.2. Minimum entropy principle.

The minimum-entropy principle holds for smooth solutions with compactly supported deviations when the stated flux conditions hold; nonnegative a and d and G(∇su):∇u ≥ 0 are essential conditions.The paper also states that positive internal energy follows from positive density and the minimum entropy principle.Source: §3 · p. 6, §3.2 · pp. 8, 10

4. Entropy inequalities.

Section 4 examines whether the regularization satisfies generalized entropy inequalities and establishes that, under the stated assumptions, this holds for all generalized entropies exactly when a = d.

4.1. Generalized entropies.

For weak solutions, the entropy inequality holds for all generalized entropies ρf(s) if and only if a = d.Source: §4.1 · p. 11
When a ≠ d, a tailored generalized entropy and initial state yield a violation of the entropy inequality near the initial time.Source: §4.1 · p. 12
The a = d regularization is the most dissipative in the family and is proposed as a candidate for robust first-order numerical methods.Source: §4.1 · p. 12
Corollary 4.3 gives a parameter range under which weak solutions satisfy the physical entropy inequality.Source: §4.1 · p. 13

4.2. Ideal gas.

In the ideal-gas illustration, γ is within the stated admissible range for the physical entropy inequality.Source: §4.2 · p. 13

5. Discussion.

The discussion connects the general regularization to parabolic and Navier–Stokes formulations, and concludes that choosing a = d gives compatibility with all generalized Harten entropies while retaining the minimum entropy principle.

5.1. Parabolic regularization.

The parabolic regularization is a special case of the general family with a = d = ǫ and G = ǫρ∇u.The general family also decouples velocity regularization from density and internal-energy regularization, allowing G to depend on the symmetric gradient for rotation invariance.Source: §5.1 · pp. 13–14

5.2. Connection with phenomenological models.

The regularized system’s heat-flux term reduces to Fourier’s law when a = d and de = cv dT.In that case, l − ef = d(ρ, e)ρcv∇T.Source: §5.2 · p. 14
The authors derive conservation equations resembling Brenner’s model and a two-velocity formulation without invoking an ad hoc phenomenological assumption.They present this mathematical connection while emphasizing their aim of a numerical device satisfying positivity and entropy properties.Source: §5.2 · p. 14

5.3. Conclusions.

Choosing a = d gives the most robust regularization, compatible with all generalized entropies of Harten et al..The paper states that the minimum entropy principle requires the specified form of q in (5.11).Source: §5.3 · p. 15

A.1. Chain rule.

The subsection introduces a C1 diffeomorphism from pairs (α, β) to pairs (φ, ψ).

A.1. Chain rule.

Φ maps (α, β) to (φ(α, β), ψ(α, β)).The map is specified as a C1 diffeomorphism Φ: R2 → R2.Source: §A.1 · p. 15
The map’s input is a pair of real variables, α and β.Its domain is R2.Source: §A.1 · p. 15
The map’s output is a pair of real-valued components, φ and ψ.Its codomain is R2.Source: §A.1 · p. 15

A.2. Speed of sound.

The appendix defines the squared speed of sound through pressure as a function of density and specific entropy, then relates that definition through the chain rule.

A.2. Speed of sound.

The squared speed of sound is defined through the pressure function of density and specific entropy.The passage identifies c2 as the partial derivative of pressure with respect to density at fixed specific entropy.Source: §A.2 · p. 15
The chain rule is used to express the speed-of-sound definition in an equivalent form.The appendix then applies the variable assignment α = ρ, β = e, and ψ = s.Source: §A.2 · p. 15
The appendix uses representations of pe and pρ to obtain a further expression for the speed of sound.The supplied passage does not state that expression.Source: §A.2 · p. 15

A.3. Convexity of the entropy, det(Σ).

The appendix relates a matrix associated with entropy curvature to convexity and the determinant of Σ.

A.3. Convexity of the entropy, det(Σ).

The defined matrix is, up to a factor of ρ, the entropy Hessian in variables (ρ−1, e).The passage states that entropy convexity implies see and ρ−1∂ρ(ρ2sρ) are negative.Source: §A.3 · p. 16
The entropy convexity assumption implies negativity of see and ρ−1∂ρ(ρ2sρ).Source: §A.3 · p. 16
The subsection establishes a characterization of det(Σ).The supplied passages do not state the characterization’s formula.Source: §A.3 · p. 16

A.4. Specific heat at constant pressure .

This appendix defines specific heat at constant pressure and derives its positivity from thermodynamic derivative relations. Under the stated convexity and temperature assumptions, the Euler system is hyperbolic without requiring positive pressure.

A.4. Specific heat at constant pressure .

cp is defined as specific heat at constant pressure, cp(ρ, e) = T ∂Ts(T, p).Source: §A.4 · p. 16
The chain-rule relation 1 = T se = T sp pe + T sT Te reduces the positivity proof to showing sp pe < 0.The appendix concludes sp(p, T)pe(ρ, e) < 0 using the determinant and assumptions (2.8) and (2.10).Source: §A.4 · pp. 16–17
The Euler system is hyperbolic under convexity assumption (2.8) and positive temperature (2.10); positive pressure is not required.The passage links convexity to Te > 0, then to cp > 0 and c2 > 0.Source: §A.4 · p. 17

A.5. Matrix H2.

The appendix collects three properties of H2 used in entropy-inequality investigations: zero determinant, negative semidefiniteness, and a derivative identity. It relates the semidefiniteness result to Harten et al. [10, p. 2126].

A.5. Matrix H2.

H2 is introduced as the matrix whose induced quadratic form appears in investigations of entropy inequalities.Source: §A.5 · p. 17
H2 has zero determinant, is negative semidefinite, and satisfies h22pρ − h12pe = 0.The proof establishes negative semidefiniteness using 1 < cpTe and P + see < 0.Source: §A.5 · p. 17
The appendix says the zero-determinant result is essentially the result established in Harten et al. [10, p. 2126].Source: §A.5 · p. 17