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MUSCL and WENO reconstruct a translating affine profile exactly

Example

Let $I_{i} = [i h, (i + 1) h)$ , $x_{i} = (i + \frac{1}{2}) h$ , and $u (x, t) = x - a t$ for a fixed $a \in R$ . On this affine profile, both the minmod MUSCL reconstruction and each of the three normalized WENO5 reconstructions give the exact upwind face value; the corresponding conservative forward-Euler update advances every cell average exactly by one time step.

Facts & Assumptions

Given: $h > 0$ , $Δ t > 0$ , uniform cells $I_{i}$ , fixed advection speed $a$ , the profile $u (x, t) = x - a t$ , and the reconstruction conventions of the A page.

[F1]

Finite-volume MUSCL, nodal ENO2 and WENO5 reconstruction conventions: cell averages are reconstructed by the displayed MUSCL slope and WENO candidate functionals, normalized weights sum to one, and positive/negative advection selects the appropriate upwind trace.

[F2]

The WENO5 ideal linear weights reproduce quartics at the right face: each WENO candidate functional is exact on polynomials of degree at most two; in particular all three candidate face values agree with the exact face value for affine data.

[F3]

MUSCL/minmod face states are second order on smooth monotone data: on a smooth strictly monotone profile, minmod selects the common-sign one-sided differences and the face states have second-order accuracy.

[F4]

Fixed-epsilon WENO-JS and WENO-Z face reconstructions are fifth order: for C5 data and fixed positive $ϵ_{0}$ , the displayed WENO-JS and WENO-Z reconstructions have fifth-order face accuracy.

Verification

1.1givenF1algebra

Since the average of an affine function over a cell equals its value at the cell center, $U_{i}^{n} = x_{i} - a t_{n}$ . Hence $U_{i}^{n} - U_{i - 1}^{n} = U_{i + 1}^{n} - U_{i}^{n} = h$ and the minmod slope is $s_{i} = 1$ . The right MUSCL trace is $U_{i}^{n} + h / 2 = (i + 1) h - a t_{n} = u (x_{i + 1 / 2}, t_{n})$ , with the reflected left trace giving the same exact value at the face.

1.2F1F2F3F4algebra

The WENO candidates reproduce affine polynomials by [F2], so $p_{0} = p_{1} = p_{2} = u (x_{i + 1 / 2}, t_{n})$ . Each normalized weight triple therefore returns this same exact value. The uniform smooth profile also satisfies [F3] and [F4], with zero error here because it is affine.

2.1F1step 1.1step 1.2algebra∎

For $a > 0$ , the upwind flux is $a ((i + 1) h - a t_{n})$ and the left face flux is $a (i h - a t_{n})$ ; their difference is $a h$ , so the update gives $U_{i}^{n + 1} = x_{i} - a t_{n} - a Δ t$ , the exact average at $t_{n + 1}$ . For $a < 0$ the reflected right trace yields the same interface value and flux difference; for $a = 0$ both the flux and time change vanish.

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