TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Direct forward-Euler MUSCL/minmod is TVD under the sharp two-thirds CFL bound

Statement

Let $h > 0$ , $Δ t \geq 0$ , $a \in R$ , and $μ = ∣ a ∣ Δ t / h$ . Use the direct reconstructed-upwind forward-Euler update for scalar linear advection from Finite-volume MUSCL, nodal ENO2 and WENO5 reconstruction conventions, with the stated MUSCL/minmod reconstruction and no time predictor; write $U^{+}$ for the sequence after one such update. The data are either a real bi-infinite sequence with finite discrete total variation $TV (U) = \sum_{i \in Z} ∣ U_{i} - U_{i - 1} ∣ < \infty,$ or a real $N$ -periodic sequence, for an integer $N \geq 1$ , with ${TV}_{N} (U) = \sum_{i = 0}^{N - 1} ∣ U_{i} - U_{i - 1} ∣,$ where periodic indices are taken modulo $N$ . If $0 \leq μ \leq 2 / 3$ , one update satisfies $TV (U^{+}) \leq TV (U)$ in the bi-infinite case and ${TV}_{N} (U^{+}) \leq {TV}_{N} (U)$ in the periodic case.

The constant $2 / 3$ is sharp within the original range $0 \leq μ \leq 1$ : for every $2 / 3 < μ \leq 1$ there are finite-support data of variation $2$ whose updated variation is $3 μ > 2$ . For every period $N \geq 7$ , the same witness periodized with period $N$ has the same strict increase.

Facts & Assumptions

Given: $h > 0$ , $Δ t \geq 0$ , $a \in R$ , the direct forward-Euler MUSCL/minmod update for scalar linear advection, and either a bi-infinite real sequence of finite discrete total variation or an $N$ -periodic real sequence with $N \geq 1$ .

[F1]

Finite-volume MUSCL, nodal ENO2 and WENO5 reconstruction conventions: the MUSCL/minmod slope is $s_{i} = h^{- 1} minmod (U_{i} - U_{i - 1}, U_{i + 1} - U_{i})$ , with face states $U_{i} \pm h s_{i} / 2$ . Minmod vanishes when either argument is zero or their signs differ, and otherwise has their common sign and magnitude no larger than either argument's magnitude.

[F2]

Finite-volume MUSCL, nodal ENO2 and WENO5 reconstruction conventions: for $a > 0$ , the direct update is $U_{i}^{+} = U_{i} - μ [d_{i} + (m_{i} - m_{i - 1}) / 2]$ , where $d_{i} = U_{i} - U_{i - 1}$ and $m_{i} = h s_{i}$ . For $a < 0$ , the fluxes are $F_{i + 1 / 2} = a (U_{i + 1} - m_{i + 1} / 2)$ and $F_{i - 1 / 2} = a (U_{i} - m_{i} / 2)$ .

Proof

1.1givenF1F2algebra

First suppose $a > 0$ . If $d_{i} \neq 0$ , then $m_{i} / d_{i}$ and $m_{i - 1} / d_{i}$ lie in $[0, 1]$ : in each minmod, $d_{i}$ is one argument, so a nonzero output has the sign of $d_{i}$ and magnitude at most $∣ d_{i} ∣$ . Set $C_{i} = μ [1 + \frac{m_{i} / d_{i} - m_{i - 1} / d_{i}}{2}]$ in this case. If $d_{i} = 0$ , then $m_{i} = m_{i - 1} = 0$ ; set $C_{i} = μ$ . In either case $μ / 2 \leq C_{i} \leq 3 μ / 2$ , and the exact update satisfies $U_{i}^{+} = U_{i} - C_{i} d_{i} = (1 - C_{i}) U_{i} + C_{i} U_{i - 1}$ . Thus $0 \leq C_{i} \leq 1$ when $μ \leq 2 / 3$ , including equality.

2.1step 1.1algebra

Subtract the identity for $U_{i - 1}^{+}$ in step 1.1 from that for $U_{i}^{+}$ to obtain $d_{i}^{+} = (1 - C_{i}) d_{i} + C_{i - 1} d_{i - 1}$ . The coefficients are nonnegative, so $∣ d_{i}^{+} ∣ \leq (1 - C_{i}) ∣ d_{i} ∣ + C_{i - 1} ∣ d_{i - 1} ∣$ . For periodic data, $C_{i}$ is $N$ -periodic because it is defined from shifts of $U$ ; summing over one period and cyclically reindexing gives ${TV}_{N} (U^{+}) \leq \sum_{i = 0}^{N - 1} [(1 - C_{i}) + C_{i}] ∣ d_{i} ∣ = {TV}_{N} (U)$ . This also covers $N = 1$ , when every $d_{i} = 0$ . For bi-infinite data, both nonnegative series on the right are bounded by $\sum_{i} ∣ d_{i} ∣ < \infty$ . Reindexing the absolutely convergent second series gives $TV (U^{+}) \leq \sum_{i} [(1 - C_{i}) + C_{i}] ∣ d_{i} ∣ = TV (U)$ , so the updated sequence still has finite variation.

3.1F1F2step 2.1algebra

Now suppose $a < 0$ . At face $i + 1 / 2$ , [F2] gives $F_{i + 1 / 2} = a (U_{i + 1} - m_{i + 1} / 2)$ and $F_{i - 1 / 2} = a (U_{i} - m_{i} / 2)$ , hence $U_{i}^{+} = U_{i} + μ [(U_{i + 1} - U_{i}) - (m_{i + 1} - m_{i}) / 2]$ . Reflect the data by $V_{j} = U_{- j}$ . Its minmod increments obey $m_{j}^{V} = - m_{- j}$ , because reflection reverses each adjacent difference and minmod is odd under simultaneous sign reversal. With $i = - j$ , the positive-speed expression for $V_{j}$ is $V_{j} - μ [(V_{j} - V_{j - 1}) + (m_{j}^{V} - m_{j - 1}^{V}) / 2] = U_{i} + μ [(U_{i + 1} - U_{i}) - (m_{i + 1} - m_{i}) / 2]$ , exactly the negative-speed update just computed. Also $V_{j} - V_{j - 1} = - (U_{- j + 1} - U_{- j})$ , so reflection only reindexes the absolute increments and preserves both the bi-infinite and periodic variation sums. Step 2.1 therefore proves the claim for negative velocity. If $a = 0$ , both fluxes vanish; if $Δ t = 0$ , their difference is multiplied by zero in the update. In either case the update is the identity, covering $μ = 0$ .

4.1F1F2step 2.1step 3.1algebra∎

Fix $2 / 3 < μ \leq 1$ , any $h > 0$ and any $a \neq 0$ , and take $Δ t = μ h / ∣ a ∣$ . For positive velocity choose $U_{- 1} = U_{1} = 1 / 2$ , $U_{0} = 1$ , and $U_{i} = 0$ otherwise; its initial variation is $∣ 1 / 2 ∣ + ∣ 1 / 2 ∣ + ∣ - 1 / 2 ∣ + ∣ - 1 / 2 ∣ = 2$ . The only nonzero differences are $d_{- 1} = d_{0} = 1 / 2$ and $d_{1} = d_{2} = - 1 / 2$ , so the only nonzero minmod increments are $m_{- 1} = 1 / 2$ and $m_{1} = - 1 / 2$ . Substituting these in [F2], the four affected updates are $U_{- 1}^{+} = 1 / 2 - μ (1 / 2 + 1 / 4) = A$ , $U_{0}^{+} = 1 - μ (1 / 2 - 1 / 4) = B$ , $U_{1}^{+} = 1 / 2 - μ (- 1 / 2 - 1 / 4) = C$ , and $U_{2}^{+} = 0 - μ (- 1 / 2 + 1 / 4) = D$ , where $A = 1 / 2 - 3 μ / 4$ , $B = 1 - μ / 4$ , $C = 1 / 2 + 3 μ / 4$ , and $D = μ / 4$ ; every other update is zero because its state and all adjacent minmod increments are zero. Here $A < 0$ , $B \geq 3 / 4 > 0$ , $C - B = μ - 1 / 2 > 0$ , $D > 0$ , and $C - D = 1 / 2 + μ / 2 > 0$ , so $A < 0 < B < C$ and $0 < D < C$ . Hence $TV (U^{+}) = - A + (B - A) + (C - B) + (C - D) + D = 2 (C - A) = 3 μ > 2$ . For negative velocity reflect this witness; step 3.1 preserves its updated variation. Periodizing the data with any $N \geq 7$ leaves at least four zero initial cells between successive three-cell supports and at least three zero updated entries between the four affected entries; since a positive-velocity update uses only cells $i - 2, \dots, i + 1$ and a negative-velocity update uses only cells $i - 1, \dots, i + 2$ , no stencil meets two distinct support copies, and the periodic variation is the same $3 μ$ . At $μ = 2 / 3$ , the coefficient argument in steps 1.1–2.1 still proves TVD, and the witness updates to $(0, 5 / 6, 1, 1 / 6)$ with variation $2$ ; every larger CFL number through one fails.

Depends on

Finite-volume MUSCL, nodal ENO2 and WENO5 reconstruction conventions

Used by

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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)