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Burgers Godunov flux under the invariant-interval CFL bound

Example

Assume ${A C}_{ω}$ , and also assume Dependent Choice $D C$ if $u_{0} \in L^{1} (R)$ . Let $f_{B} (s) = s^{2} / 2$ and use the Godunov flux $G$ of Godunov numerical flux for Burgers' equation. For every compact invariant interval $I \subseteq [- M, M]$ with $M > 0$ , the two coordinate Lipschitz bounds can be taken to be $L_{1} = L_{2} = M$ , and $G$ is nondecreasing in its first argument and nonincreasing in its second. Therefore, for any $u_{0} \in {B V}_{b} (R)$ with values in $I$ , every mesh family satisfying $\frac{τ_{m}}{h_{m}} \leq \frac{1}{2 M}, \frac{τ_{m}}{h_{m}} \geq c_{*} > 0, h_{m}, τ_{m} \to 0$ falls under Strong entropy convergence of monotone finite-volume schemes and converges strongly locally to the unique Burgers entropy solution.

Facts & Assumptions

Given: ${A C}_{ω}$ , $f_{B} (s) = s^{2} / 2$ , a compact interval $I \subseteq [- M, M]$ with $M > 0$ , bounded BV data $u_{0}$ in $I$ , the Godunov flux $G$ , and $λ = τ / h$ ; additionally assume $D C$ when $u_{0} \in L^{1} (R)$ .

[F1]

The Godunov flux has the listed increasing-state and decreasing-state branches (Godunov numerical flux for Burgers' equation).

[F2]

$G$ is continuous and consistent: $G (a, a) = f_{B} (a)$ (The Burgers Godunov flux is continuous and consistent).

[F3]

Data in ${B V}_{b} (R)$ have finite translation variation and are locally integrable equivalence classes (Bounded BV data on the real line).

[F4]

Under ${A C}_{ω}$ , and with DC additionally when $u_{0} \in L^{1} (R)$ , if $F$ is locally Lipschitz, consistent and coordinatewise monotone on the data range, and $λ (L_{1} + L_{2}) \leq 1$ with positive lower $λ$ , the corresponding finite-volume reconstructions converge strongly locally to the unique entropy solution (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain, Strong entropy convergence of monotone finite-volume schemes).

Verification

1.1F1F2givenalgebra

Fix $b \in [- M, M]$ . On each branch in [F1], the function $a \mapsto G (a, b)$ is either constant, $a^{2} / 2$ , or $\max {a^{2}, b^{2}} / 2$ . It is nondecreasing: when $a \leq b$ , the active values are $b^{2} / 2$ on the negative-negative branch, $0$ on the crossing branch, and $a^{2} / 2$ on the nonnegative branch; when $b < a$ , the maximum is constant while $∣ a ∣ \leq ∣ b ∣$ and increases as $a^{2} / 2$ once $a$ is the larger positive magnitude. The branch values agree at their boundaries by [F2], so the monotonicity persists across each switch.

1.2F1F2algebra

Fix $a \in [- M, M]$ . On the branches of [F1], $b \mapsto G (a, b)$ is either constant, $b^{2} / 2$ for $b \leq 0$ , or $\max {a^{2}, b^{2}} / 2$ on $b < a$ . As $b$ increases, $b^{2} / 2$ decreases on the negative axis; on $b < a$ , the maximum decreases while $∣ b ∣ > ∣ a ∣$ and is constant after $∣ b ∣ \leq ∣ a ∣$ ; the remaining $a \leq b$ branches are constant or decrease to zero. Boundary values agree by [F2]. Hence $G (a, \cdot)$ is nonincreasing.

1.3F1F2algebra

On every smooth branch the derivative in the varying coordinate has absolute value at most $R$ on $[- R, R]^{2}$ . The maximum branch has the same bound because $\max {a^{2}, b^{2}} / 2$ is piecewise one of the two square branches. The finite branch partition and continuity from [F2] therefore give coordinate Lipschitz bounds $R$ on every compact square $[- R, R]^{2}$ , proving that $G$ is locally Lipschitz on $R^{2}$ . Applying the same estimate with $R = M$ gives coordinate Lipschitz bounds $L_{1} = L_{2} = M$ on the target square. Consistency is [F2], so the full CFL bound is $λ (L_{1} + L_{2}) = 2 M λ \leq 1$ .

2.1F3F4step 1.1step 1.2step 1.3given∎

The hypotheses of [F4] now hold for every stated family: the data are bounded BV by [F3], the flux is locally Lipschitz, monotone and consistent by steps 1.1–1.3, the full CFL bound holds, and the mesh ratio has the required positive lower bound. Thus the reconstructions converge strongly in local $L^{1}$ to the unique entropy solution.

Depends on

Used by

Godunov scheme captures a stationary Burgers entropy shock Example

Dependency tree · two levels

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Sources

Jérôme Droniou, A beginner’s course in finite volume approximation of scalar conservation laws, complete lecture note (standard reference, not scraped)
Siddhartha Mishra, Ulrik S. Fjordholm and Rémi Abgrall, Numerical Methods for Conservation Laws, ETH lecture notes (standard reference, not scraped)