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A Roe jump property does not imply entropy admissibility

Statement refuted

For a scalar conservation law, the Roe exact flux-jump identity by itself guarantees entropy admissibility.

Facts & Assumptions

Given: Burgers' scalar conservation law, its Roe matrix, the Riemann step data $u_{0} (x) = - 1$ for $x < 0$ and $u_{0} (x) = 1$ for $x > 0$ , and the convex entropy pair below.

[F1]

Roe linearization and Roe numerical flux: the Roe jump property is the identity $\tilde{a} (u_{L}, u_{R}) (u_{R} - u_{L}) = f (u_{R}) - f (u_{L})$ , which does not itself impose an entropy inequality.

[F2]

Convex entropy admissibility for a scalar conservation law: for an entropy pair $(η, q)$ , a moving jump of speed $s$ is entropy-admissible only if $[q] - s [η] \leq 0$ .

[F3]

Rankine–Hugoniot jumps and Lax shocks: a Rankine–Hugoniot jump of speed $s$ is defined by $[f] = s [u]$ .

[F4]

A moving jump is a weak solution exactly under Rankine–Hugoniot: a Rankine–Hugoniot jump with the corresponding Riemann step data is a distributional weak solution and has the required local $L^{1}$ initial trace.

[A1]

For Burgers' flux, $η (u) = u^{2} / 2$ and $q (u) = u^{3} / 3$ satisfy $η^{''} = 1 > 0$ and $q^{'} = η^{'} f^{'}$ .

Counterexample

1.1givenF1F3F4algebra

For Burgers' flux $f (u) = u^{2} / 2$ with $u_{L} = - 1, u_{R} = 1$ , the Roe scalar $\tilde{a} (u_{L}, u_{R}) = (u_{L} + u_{R}) / 2$ is continuous at each diagonal pair, has equal-state value $\tilde{a} (u, u) = u = f^{'} (u)$ , and satisfies the jump identity for every pair because $f (u_{R}) - f (u_{L}) = \frac{1}{2} (u_{L} + u_{R}) (u_{R} - u_{L})$ . At $(- 1, 1)$ the speed is zero and $f (1) - f (- 1) = 0 = s (u_{R} - u_{L})$ , so the stationary jump satisfies the Roe identity [F1] and, by [F4], is a weak solution with the Riemann data in the Given clause.

2.1givenA1F2step 1.1algebra∎

For [A1], the entropy production coefficient is $[q] - s [η] = (1 / 3 - (- 1 / 3)) - 0 \cdot (1 / 2 - 1 / 2) = 2 / 3 > 0$ . This violates the necessary entropy inequality [F2]. Thus the Roe jump property and weak Rankine–Hugoniot balance do not guarantee entropy admissibility.

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Sources

Randall J. LeVeque, Numerical Methods for Conservation Laws (standard reference, not scraped)