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The specified direct MUSCL/minmod flux update is not TVD at CFL one

Statement refuted

The direct reconstructed-upwind forward-Euler update in Finite-volume MUSCL, nodal ENO2 and WENO5 reconstruction conventions is TVD for every scalar grid sequence whenever $∣ a ∣ Δ t / h \leq 1$ .

Facts & Assumptions

Given: $a = h = Δ t = 1$ , so the CFL number is $1$ ; the minmod slope and direct forward-Euler flux update of Finite-volume MUSCL, nodal ENO2 and WENO5 reconstruction conventions. For a finite-support sequence, $TV (U) = \sum_{i} ∣ U_{i + 1} - U_{i} ∣$ .

[F1]

Finite-volume MUSCL, nodal ENO2 and WENO5 reconstruction conventions: the slope increments are $σ_{i} = h s_{i} = minmod (U_{i} - U_{i - 1}, U_{i + 1} - U_{i})$ and the positive-velocity face flux is $F_{i + 1 / 2} = U_{i} + σ_{i} / 2$ .

Counterexample

1.1givenF1algebra

Set $U_{- 3} = U_{3} = 0$ , $U_{- 2} = U_{2} = 0$ , $U_{- 1} = U_{1} = 1 / 2$ , $U_{0} = 1$ , and $U_{i} = 0$ for all other indices. The initial variation is $1 / 2 + 1 / 2 + 1 / 2 + 1 / 2 = 2$ . By [F1], the nonzero slope increments are $σ_{- 1} = 1 / 2$ and $σ_{1} = - 1 / 2$ ; the central slope is $σ_{0} = minmod (1 / 2, - 1 / 2) = 0$ , and the adjacent outer slopes are zero.

2.1F1step 1.1algebra

The relevant right-going face fluxes from [F1] are $F_{- 3 / 2} = 0$ , $F_{- 1 / 2} = 3 / 4$ , $F_{1 / 2} = 1$ , $F_{3 / 2} = 1 / 4$ , and $F_{5 / 2} = 0$ . Substituting in the stated update gives $U_{- 1}^{n + 1} = - 1 / 4$ , $U_{0}^{n + 1} = 3 / 4$ , $U_{1}^{n + 1} = 5 / 4$ , and $U_{2}^{n + 1} = 1 / 4$ , with neighboring states zero.

3.1step 1.1step 2.1algebra∎

The new variation is $1 / 4 + 1 + 1 / 2 + 1 + 1 / 4 = 3$ , which exceeds the initial value $2$ . The data satisfy the claimed CFL bound with equality, so the proposed all-data TVD claim is false.

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)