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Exact Tadmor entropy residual for a dissipative Burgers flux

Example

For Burgers' flux $f (u) = u^{2} / 2$ and entropy $η (u) = u^{2} / 2$ , the entropy flux is $q (u) = u^{3} / 3$ and the entropy potential is $ψ (u) = u^{3} / 6$ . The symmetric two-point flux $f^{e c} (u_{L}, u_{R}) = \frac{u_{L}^{2} + u_{L} u_{R} + u_{R}^{2}}{6}$ is entropy-conservative. Adding unit dissipation in entropy variables gives $f^{e s} = f^{e c} - (u_{R} - u_{L}) / 2$ ; at the jump $1 \to 0$ its Tadmor entropy residual is exactly $- 1 / 2$ .

Facts & Assumptions

Given: Burgers' flux, the quadratic entropy pair, and the two-point flux formula. Verify compatibility, the entropy-conservative identity, and the dissipative residual at $(u_{L}, u_{R}) = (1, 0)$ .

[F1]

Convex entropy pairs, entropy variables, and entropy potentials for systems: an entropy pair satisfies $q^{'} = η^{'} f^{'}$ , with $v = η^{'}$ and $ψ = v f - q$ .

[F2]

Tadmor entropy-conservative two-point flux: a continuous, consistent, symmetric two-point flux is entropy-conservative when $[v] f^{e c} = [ψ]$ for every pair of states, with right-minus-left jumps.

[F3]

Entropy-stable flux from positive-semidefinite entropy-variable dissipation: $f^{e s} = f^{e c} - D [v] / 2$ for symmetric positive-semidefinite $D$ .

Verification

1.1givenF1algebra

Here $η^{''} (u) = 1$ , $η^{'} (u) = u$ , and $q^{'} (u) = u^{2} = η^{'} (u) f^{'} (u)$ , so this is a convex entropy pair. Thus $v = u$ and $ψ = u f - q = u^{3} / 6$ by [F1].

2.1givenF1F2step 1.1algebra

The displayed $f^{e c}$ is polynomial, hence continuous, is symmetric, and satisfies $f^{e c} (u, u) = u^{2} / 2 = f (u)$ . Moreover, $(u_{R} - u_{L}) f^{e c} (u_{L}, u_{R}) = \frac{u_{R}^{3} - u_{L}^{3}}{6} = ψ (u_{R}) - ψ (u_{L})$ , so [F2] applies using $ψ = u^{3} / 6$ from step 1.1.

3.1givenF2F3step 2.1algebra∎

Taking $D = 1$ in [F3], at $(1, 0)$ we have $f^{e c} = 1 / 6$ , $[v] = - 1$ , and $[ψ] = - 1 / 6$ . Therefore $[v] f^{e s} - [ψ] = (- 1) (1 / 6 + 1 / 2) - (- 1 / 6) = - 1 / 2$ . In general the same substitution gives residual $- \frac{1}{2} (u_{R} - u_{L})^{2} \leq 0$ , using the formula in step 2.1.

Depends on

Used by

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Sources

E. Tadmor, Entropy Stable Schemes for Hyperbolic Conservation Laws, Handbook of Numerical Methods for Hyperbolic Problems, Vol. 17 (standard reference, not scraped)
R. Bürger, P. E. Méndez, and C. Parés, On Entropy Stable Schemes for Degenerate Parabolic Multispecies Kinematic Flow Models (standard reference, not scraped)