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Positive-state Burgers shock and the monotone finite-volume baseline

Example

For $f (u) = u^{2} / 2$ and $u_{0} (x) = 1$ for $x < 0$ , $u_{0} (x) = 0$ for $x > 0$ , the entropy shock moves right at speed $1 / 2$ . For states $a, b \in [0, 1]$ , define the positive-state Burgers Godunov flux by $F_{G} (a, b) = \min_{v \in [a, b]} f (v)$ when $a \leq b$ and $F_{G} (a, b) = \max_{v \in [b, a]} f (v)$ when $a > b$ . Since $f$ is increasing on $[0, 1]$ , both cases give $F_{G} (a, b) = a^{2} / 2$ . Minmod slopes vanish for the initial step, so the first MUSCL flux update is the first-order Godunov update; the batch-29 monotone finite-volume theorem applies to its mesh-refined baseline under Countable Choice and the stated CFL conditions.

Facts & Assumptions

Given: $T > 0$ , $f (u) = u^{2} / 2$ , $u_{0} = 1_{(- \infty, 0)}$ , cells $I_{j} = [j h, (j + 1) h)$ , $h_{m} = 1 / m$ , $τ_{m} = h_{m} / 2$ , and the displayed MUSCL/minmod reconstruction. The positive-state Godunov flux is defined by the two min/max formulas in the Example.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): ${A C}_{ω}$ asserts that every countable family of nonempty sets has a choice function.

[F1]

Bounded BV data on the real line: ${B V}_{b} (R)$ consists of bounded locally integrable classes with finite translation variation; it does not require global $L^{1}$ integrability.

[F2]

Monotone finite-volume scheme and invariant-interval CFL: on an invariant interval $I$ , the shared two-point update uses a locally Lipschitz consistent flux nondecreasing in its first coordinate and nonincreasing in its second, under $λ (L_{1} + L_{2}) \leq 1$ .

[F3]

Strong entropy convergence of monotone finite-volume schemes: under ${A C}_{ω}$ , bounded BV data, a locally Lipschitz consistent coordinatewise monotone flux on the invariant interval, $h_{m}, τ_{m} \to 0$ , a positive lower bound on $τ_{m} / h_{m}$ , and the full CFL bound, the cell-average scheme converges locally to the unique bounded Kruzhkov entropy solution. Dependent Choice is additionally required only on the theorem's $u_{0} \in L^{1} (R)$ branch.

[F4]

Kruzhkov entropy solutions: a Kruzhkov solution satisfies every entropy inequality for $η_{k} (u) = ∣ u - k ∣$ , $q_{k} (u) = sgn (u - k) (f (u) - f (k))$ , and has a strong local $L^{1}$ initial trace.

[F5]

Rankine–Hugoniot jumps and Lax shocks: a jump from $u_{L}$ to $u_{R}$ with speed $s$ satisfies the weak jump law exactly when $f (u_{R}) - f (u_{L}) = s (u_{R} - u_{L})$ ; in the scalar convex case, a strict Lax shock has $f^{'} (u_{R}) < s < f^{'} (u_{L})$ .

[F6]

Finite-volume MUSCL, nodal ENO2 and WENO5 reconstruction conventions: the minmod slope is the minmod of the left and right divided differences; a positive-speed face flux uses the reconstructed trace from the cell on its left.

Verification

1.1givenA1F1F3algebra

The jump data are bounded and locally integrable. For any translation length $r > 0$ , $∣ u_{0} (x + r) - u_{0} (x) ∣$ equals $1$ on an interval of length $r$ and vanishes elsewhere, so $TV (u_{0}) = 1$ by [F1]. The data are not in global $L^{1} (R)$ , so [F3] uses only its ${A C}_{ω}$ branch and imposes no DC assumption.

1.2F2F5algebra

For $0 \leq a \leq b \leq 1$ , monotonicity of $f$ gives $\min_{v \in [a, b]} f (v) = f (a)$ ; for $0 \leq b < a \leq 1$ , it gives $\max_{v \in [b, a]} f (v) = f (a)$ . Thus the flux defined in the Example is $F_{G} (a, b) = a^{2} / 2$ on $[0, 1]^{2}$ . Use the same polynomial $F (a, b) = a^{2} / 2$ on all of $R^{2}$ for the theorem's globally defined input. Its restriction is consistent, nondecreasing in $a$ , independent of $b$ , and satisfies $∣ F (a, b) - F (a^{'}, b) ∣ \leq ∣ a - a^{'} ∣$ on $[0, 1]^{2}$ ; hence $L_{1} = 1, L_{2} = 0$ are valid in [F2]. For the particular jump $(1, 0)$ , [F5] gives speed $s = (f (1) - f (0)) / (1 - 0) = 1 / 2 > 0$ , so the interface ray lies on the left state and the flux is the physical Riemann flux.

2.1A1F2F3step 1.1step 1.2given

Here $λ_{m} = τ_{m} / h_{m} = 1 / 2$ , $h_{m}, τ_{m} \to 0$ , and $λ_{m} (L_{1} + L_{2}) = 1 / 2 \leq 1$ with $λ_{m} \geq 1 / 2$ . Therefore [F3] applies to the cell-average recurrence and gives convergence to the unique bounded entropy solution, provided the explicit shock below is identified as that solution.

3.1F3F4F5step 2.1algebra

Define $u (t, x) = 1$ for $x < t / 2$ and $u (t, x) = 0$ for $x > t / 2$ . The jump speed is $s = 1 / 2 = (f (1) - f (0)) / (1 - 0)$ , so [F5] gives the weak jump law. For $0 \leq k \leq 1$ , the Kruzhkov entropy-flux jump is $q_{k} (0) - q_{k} (1) - \frac{1}{2} (∣ 0 - k ∣ - ∣ 1 - k ∣) = k^{2} - k \leq 0$ ; for $k \leq 0$ and $k \geq 1$ direct substitution gives zero. The initial trace is strong locally because the shock moves a distance $t / 2 \to 0$ . Hence [F4] identifies this profile as a bounded Kruzhkov solution; uniqueness in [F3] makes it the convergence target from step 2.1.

4.1F2F6step 1.2algebra∎

All cell averages are constant on each side of the initial interface, and both neighboring differences change sign or vanish at the two cells adjacent to it; [F6] therefore gives zero minmod slope in every cell. At the first step $F (1, 0) = F (1, 1) = 1 / 2$ and $F (0, 0) = 0$ , so the cell to the right of the interface becomes $0 - (1 / 2) (0 - 1 / 2) = 1 / 4$ , while the cell to the left remains $1$ . The exact shock at time $τ = h / 2$ has advanced by $h / 4$ , so its average on the first right cell is also $1 / 4$ .

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Sources

Jérôme Droniou, A Beginner’s Course in Finite Volume Approximation of Scalar Conservation Laws, complete lecture notes (standard reference, not scraped)