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A first-order critical point exposes WENO-JS epsilon scaling

Example

Work in nondimensional coordinates with $u (x) = x^{2} + x^{3}$ and reconstruct the face $x = 0$ , the interface between cells $I_{- 1}$ and $I_{0}$ . This is a first-order critical point: $u^{'} (0) = 0$ and $u^{''} (0) = 2$ . If the WENO-JS parameter is scaled as $ϵ_{h} = h^{4}$ , the cell-average face state has error $\frac{54}{95} h^{4} + O (h^{5})$ , so this branch is fourth order at the face. With the fixed $ϵ_{0} > 0$ of Fixed-epsilon WENO-JS and WENO-Z face reconstructions are fifth order, the theorem's fifth-order branch applies; with the same $ϵ_{h}$ , the WENO-Z comparison has error $O (h^{6})$ on this polynomial.

Facts & Assumptions

Given: Uniform cells $I_{j} = [j h, (j + 1) h)$ , $u (x) = x^{2} + x^{3}$ , target face $x = 0$ , the displayed WENO-JS/WENO-Z weights, and $ϵ_{h} = h^{4} > 0$ .

[F1]

Finite-volume MUSCL, nodal ENO2 and WENO5 reconstruction conventions: the three cell-average candidates are $p_{0}, p_{1}, p_{2}$ , the ideal weights are $(1 / 10, 6 / 10, 3 / 10)$ , and the JS and Z weights use the displayed $β_{k}$ and $τ_{5} = ∣ β_{0} - β_{2} ∣$ .

[F2]

The WENO5 ideal linear weights reproduce quartics at the right face: the ideal linear combination reproduces cubics and each candidate is exact through degree two.

[F3]

Fixed-epsilon WENO-JS and WENO-Z face reconstructions are fifth order: fifth-order face accuracy is proved for fixed positive $ϵ_{0}$ ; this does not assert the same order for $ϵ_{h} = h^{4}$ .

Verification

1.1givenF1F2algebra

The five stencil averages, ordered from $i - 2$ through $i + 2$ with $i = - 1$ , are $(U_{i - 2}, U_{i - 1}, U_{i}, U_{i + 1}, U_{i + 2}) = h^{2} (19 / 3, 7 / 3, 1 / 3, 1 / 3, 7 / 3) + h^{3} (- 65 / 4, - 15 / 4, - 1 / 4, 1 / 4, 15 / 4)$ . Substitution in the three candidate formulas gives $(p_{0}, p_{1}, p_{2}) = h^{3} (- 3 / 2, 1 / 2, - 1 / 2)$ ; the quadratic contributions vanish by [F2].

2.1F1step 1.1algebra

Substitution of the same averages in the indicator formulas gives exactly $β_{0} = (16 / 3) h^{4} - 37 h^{5} + (355 / 4) h^{6}$ , $β_{1} = (16 / 3) h^{4} - 17 h^{5} + (55 / 4) h^{6}$ , and $β_{2} = (16 / 3) h^{4} + 15 h^{5} + (43 / 4) h^{6}$ . Put $C = 19 / 3$ and $(c_{0}, c_{1}, c_{2}) = (- 37, - 17, 15)$ ; then $ϵ_{h} + β_{k} = C h^{4} + c_{k} h^{5} + O (h^{6})$ .

3.1F1F2step 1.1step 2.1algebra

Expanding $α_{k}^{J} = d_{k} (C h^{4} + c_{k} h^{5} + O (h^{6}))^{- 2}$ and normalizing gives $ω_{k}^{J} = d_{k} [1 - 2 (c_{k} - \overset{ˉ}{c}) h / C + O (h^{2})]$ , where $\overset{ˉ}{c} = \sum d_{k} c_{k} = - 47 / 5$ . Since $\sum d_{k} p_{k} = 0$ by [F2], and $\sum_{k} d_{k} (c_{k} - \overset{ˉ}{c}) (- 3 / 2, 1 / 2, - 1 / 2)_{k} = - 9 / 5$ , the reconstructed state is $\sum_{k} ω_{k}^{J} p_{k} = - \frac{2 h^{4}}{C} (- 9 / 5) + O (h^{5}) = \frac{54}{95} h^{4} + O (h^{5})$ . Thus the face error is not $O (h^{5})$ .

4.1F1F2F3step 1.1step 2.1step 3.1algebra∎

By step 3.1 the scaled-epsilon JS state has a nonzero $h^{4}$ leading error. For WENO-Z, $τ_{5} = ∣ - 52 h^{5} + 78 h^{6} ∣ = 52 h^{5} + O (h^{6})$ , so $τ_{5} / (ϵ_{h} + β_{k}) = (52 / C) h + O (h^{2})$ and its square has the same leading $O (h^{2})$ term for each $k$ . That common term cancels on normalization, giving $ω_{k}^{Z} - d_{k} = O (h^{3})$ . Since the ideal combination is exactly zero on this cubic, [F2] gives $\sum_{k} ω_{k}^{Z} p_{k} = O (h^{6})$ . This establishes the stated scaled-epsilon contrast; the fixed- $ϵ_{0}$ theorem in [F3] remains a separate branch.

Depends on

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Sources

Andrew K. Henrick, Tariq D. Aslam and Joseph M. Powers, Mapped weighted essentially non-oscillatory schemes: Achieving optimal order near critical points (standard reference, not scraped)
Guang-Shan Jiang and Chi-Wang Shu, Efficient Implementation of Weighted ENO Schemes (standard reference, not scraped)