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The WENO5 ideal linear weights reproduce quartics at the right face

Statement

Let $h > 0$ , let $I_{j} = [j h, (j + 1) h)$ , and let $x_{i + 1 / 2} = (i + 1) h$ . For $u \in C^{5}$ on the union of cells $I_{i - 2}, \dots, I_{i + 2}$ , let $p_{0}, p_{1}, p_{2}$ be the cell-average functionals in Finite-volume MUSCL, nodal ENO2 and WENO5 reconstruction conventions. Each candidate reproduces every polynomial of degree at most two exactly at $x_{i + 1 / 2}$ . The ideal combination $p_{o p t} = p_{0} / 10 + 3 p_{1} / 5 + 3 p_{2} / 10$ reproduces every polynomial of degree at most four and satisfies $∣ p_{o p t} - u (x_{i + 1 / 2}) ∣ \leq (23 / 15) (3^{5} / 5!) \sup ∣ u^{(5)} ∣ h^{5}$ . Each candidate separately satisfies $∣ p_{k} - u (x_{i + 1 / 2}) ∣ \leq 15 \sup ∣ u^{'''} ∣ h^{3}$ .

Facts & Assumptions

Given: Uniform cell averages of a C5 function, the three candidate functionals and the ideal weights in Finite-volume MUSCL, nodal ENO2 and WENO5 reconstruction conventions.

[F1]

Finite-volume MUSCL, nodal ENO2 and WENO5 reconstruction conventions: the candidate coefficient vectors, in order $(U_{i - 2}, U_{i - 1}, U_{i}, U_{i + 1}, U_{i + 2})$ , are $(2, - 7, 11, 0, 0) / 6$ , $(0, - 1, 5, 2, 0) / 6$ , and $(0, 0, 2, 5, - 1) / 6$ ; the ideal weights are $(1 / 10, 3 / 5, 3 / 10)$ .

[F2]

A uniform derivative bound gives a uniform Taylor remainder bound: if $∣ u^{(m + 1)} ∣ \leq M$ on the interval between a center $c$ and $x$ , the degree- $m$ Taylor polynomial at $c$ has remainder bounded by $M ∣ x - c ∣^{m + 1} / (m + 1)!$ .

Proof

1.1F1algebra

Combining the three coefficient vectors in [F1] gives $p_{o p t} = (2 U_{i - 2} - 13 U_{i - 1} + 47 U_{i} + 27 U_{i + 1} - 3 U_{i + 2}) / 60$ .

1.2F1F2algebra

For $p_{0}$ , the coefficient vector $(2, - 7, 11)$ has dot products $6$ , $0$ , and $0$ with the average vectors $(1, 1, 1)$ , $(- 5 / 2, - 3 / 2, - 1 / 2)$ , and $(19 / 3, 7 / 3, 1 / 3)$ . For $p_{1}$ , $(- 1, 5, 2)$ has dot products $6$ , $0$ , and $0$ with $(1, 1, 1)$ , $(- 3 / 2, - 1 / 2, 1 / 2)$ , and $(7 / 3, 1 / 3, 1 / 3)$ . For $p_{2}$ , $(2, 5, - 1)$ has dot products $6$ , $0$ , and $0$ with $(1, 1, 1)$ , $(- 1 / 2, 1 / 2, 3 / 2)$ , and $(1 / 3, 1 / 3, 7 / 3)$ . After division by $6$ , each candidate exactly reproduces $1, z, z^{2}$ at the face. The degree-two Taylor polynomial at $x_{i + 1 / 2}$ therefore cancels from $p_{k} - u (x_{i + 1 / 2})$ . By [F2], the pointwise remainder on each stencil is bounded by $M_{3} (3 h)^{3} / 3!$ , where $M_{3} = \sup ∣ u^{'''} ∣$ ; averaging preserves this bound. The largest sum of absolute candidate coefficients is $10 / 3$ , giving $∣ p_{k} - u (x_{i + 1 / 2}) ∣ \leq (10 / 3) M_{3} (3 h)^{3} / 6 = 15 M_{3} h^{3}$ .

2.1step 1.1algebra

Write $z = (x - x_{i + 1 / 2}) / h$ . The vectors of cell averages of $1, z, z^{2}, z^{3}, z^{4}$ over the five cells, ordered from $i - 2$ through $i + 2$ , are respectively $(1, 1, 1, 1, 1)$ , $(- 5 / 2, - 3 / 2, - 1 / 2, 1 / 2, 3 / 2)$ , $(19 / 3, 7 / 3, 1 / 3, 1 / 3, 7 / 3)$ , $(- 65 / 4, - 15 / 4, - 1 / 4, 1 / 4, 15 / 4)$ , and $(211 / 5, 31 / 5, 1 / 5, 1 / 5, 31 / 5)$ . Direct multiplication gives dot products $60 / 60$ , $0 / 60$ , $0 / 180$ , $0 / 240$ , and $0 / 300$ with $(2, - 13, 47, 27, - 3) / 60$ , respectively. Thus the functional in step 1.1 returns the value at $z = 0$ for every polynomial of degree at most four.

3.1F1F2step 2.1algebra∎

Let $T_{4}$ be the degree-four Taylor polynomial of $u$ at $x_{i + 1 / 2}$ and put $M_{5} = \sup ∣ u^{(5)} ∣$ on the stencil. By [F2], the cell-average remainder after subtracting $T_{4}$ has magnitude at most $M_{5} (3 h)^{5} / 5!$ in each cell. Step 2.1 reproduces $T_{4}$ exactly, and the absolute coefficients of $p_{o p t}$ sum to $(2 + 13 + 47 + 27 + 3) / 60 = 23 / 15$ . Therefore $∣ p_{o p t} - u (x_{i + 1 / 2}) ∣ \leq (23 / 15) M_{5} (3 h)^{5} / 5!$ , as claimed.

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Sources

Chi-Wang Shu, Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws, complete lecture notes (standard reference, not scraped)
Guang-Shan Jiang and Chi-Wang Shu, Efficient Implementation of Weighted ENO Schemes (standard reference, not scraped)