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Entropy dissipation of the orthogonal finite-volume viscous scheme

Statement

Every differentiable solution of One-dimensional orthogonal finite-volume viscous scheme with $U_{i} (t) \in G$ satisfies

$\frac{d}{d t} \sum_{i = 1}^{N} h η (U_{i}) = - \sum_{i = 1}^{N - 1} \frac{(v_{i + 1} - v_{i})^{T} K_{i + 1 / 2} (v_{i + 1} - v_{i})}{h} \leq 0, \frac{d}{d t} \sum_{i = 1}^{N} h U_{i} = 0.$

The endpoint viscous fluxes vanish by definition, and the face quadratic forms use the state block $K_{i + 1 / 2} = K_{11} ((U_{i} + U_{i + 1}) / 2)$ .

Facts & Assumptions

Given: $L > 0$ , an integer $N \geq 1$ , $h = L / N$ , differentiable cell states $U_{i} (t) \in G$ , and the finite-volume viscous ODE in One-dimensional orthogonal finite-volume viscous scheme.

[F1]

One-dimensional orthogonal finite-volume viscous scheme: ${\dot{U}}_{i} = (G_{i + 1 / 2} - G_{i - 1 / 2}) / h$ , $G_{i + 1 / 2} = K_{i + 1 / 2} (v_{i + 1} - v_{i}) / h$ , and $G_{1 / 2} = G_{N + 1 / 2} = 0$ .

[F2]

Positive-semidefinite entropy-variable factorization of viscous flux: each one-direction state block $K_{11} (U)$ is symmetric positive semidefinite.

[F3]

Strict convexity and entropy variables for the ideal-gas Euler flux: $v (U) = \nabla_{U} η (U)$ on $G$ .

Proof

1.1F1algebra

For the total conservative state, substitute [F1] and telescope the finite sum: $\frac{d}{d t} \sum_{i = 1}^{N} h U_{i} = \sum_{i = 1}^{N} (G_{i + 1 / 2} - G_{i - 1 / 2}) = G_{N + 1 / 2} - G_{1 / 2} = 0$ .

1.2F1F3givenalgebra

Since $v_{i} = \nabla_{U} η (U_{i})$ by [F3], the chain rule gives $\frac{d}{d t} \sum_{i} h η (U_{i}) = \sum_{i} h v_{i}^{T} {\dot{U}}_{i}$ . Substituting [F1] cancels the factors $h$ and gives $\sum_{i} v_{i}^{T} (G_{i + 1 / 2} - G_{i - 1 / 2})$ .

2.1F1step 1.2algebra

At each interior face $i + 1 / 2$ , the two adjacent cells contribute $v_{i}^{T} G_{i + 1 / 2} - v_{i + 1}^{T} G_{i + 1 / 2} = - (v_{i + 1} - v_{i})^{T} G_{i + 1 / 2}$ . The boundary terms vanish by [F1]. Substituting the face flux gives the exact negative sum in the Statement.

3.1F1F2step 2.1algebra∎

Each midpoint state is admissible by the convexity built into [F1], and its matrix block is positive semidefinite by [F2]. Thus every face term $(v_{i + 1} - v_{i})^{T} K_{i + 1 / 2} (v_{i + 1} - v_{i}) / h$ is nonnegative because $h > 0$ . This proves the entropy inequality, including the case $N = 1$ when the face sum is empty.

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Sources

R. Eymard, T. Gallouët, and R. Herbin, Finite Volume Methods, January 2019 updated author text, complete manuscript (standard reference, not scraped)
Jérôme Droniou, Finite volume schemes for diffusion equations: introduction to and review of modern methods, 2014, complete preprint (standard reference, not scraped)