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A linearly-degenerate characteristic curve is a contact wave

Statement

Let $r_{i}$ be linearly degenerate on a neighborhood and let $R_{i} (σ; u_{0})$ be its local rarefaction curve. Then the states $u_{0}$ and $u_{1} = R_{i} (σ; u_{0})$ are joined by a Rankine–Hugoniot jump of speed $λ_{i} (u_{0})$ , and the corresponding moving discontinuity is a weak solution. The characteristic speed is equal on the two sides; this is a contact wave, not a strict Lax shock.

Facts & Assumptions

Given: A $C^{2}$ linearly-degenerate eigenpair and its local integral curve.

[F1]

Strictly hyperbolic conservation systems and characteristic families: $D f (u) r_{i} (u) = λ_{i} (u) r_{i} (u)$ and $D λ_{i} (u) r_{i} (u) = 0$ for a linearly-degenerate family.

[F2]

Local characteristic rarefaction curve: $R_{i}^{'} = r_{i} (R_{i})$ .

[F3]

Rankine–Hugoniot jumps and Lax shocks: the jump condition is $[f] = s [u]$ .

[F4]

A moving jump is a weak solution exactly under Rankine–Hugoniot: a moving jump is weak exactly when that equality holds.

Proof

1.1givenF1F2algebra

Along the curve, the chain rule gives $\frac{d}{d σ} λ_{i} (R_{i} (σ; u_{0})) = D λ_{i} (R_{i}) r_{i} (R_{i}) = 0$ , so $λ_{i} (R_{i}) = λ_{i} (u_{0})$ . Also $\frac{d}{d σ} f (R_{i}) = D f (R_{i}) r_{i} (R_{i}) = λ_{i} (u_{0}) r_{i} (R_{i})$ .

2.1step 1.1F3F4algebra∎

Integrating the second identity from $0$ to $σ$ yields $f (u_{1}) - f (u_{0}) = λ_{i} (u_{0}) (u_{1} - u_{0})$ , the Rankine–Hugoniot equality [F3]. Applying [F4] gives the weak moving contact wave; equality of the characteristic speed on both sides was proved in step 1.1.

Depends on

Used by

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Alberto Bressan, Hyperbolic Conservation Laws: An Illustrated Tutorial (standard reference, not scraped)
Randall J. LeVeque, Numerical Methods for Conservation Laws (standard reference, not scraped)