An exact optimal cell average decomposition for ℙ⁸ and ℙ⁹

Abstract

Cui, Ding and Wu introduced the optimal cell average decomposition (OCAD) problem, whose solution is exactly the largest bound-preserving CFL number of a Zhang–Shu limited high-order scheme on a Cartesian cell. They solved it for the tensor-product spaces ℚᵏ for every k and for ℙᵏ with 1 ≤ k ≤ 7, and conjectured that two identities — a strong duality and the existence of a critical positive polynomial that is a square — persist for all k; for ℙᵏ with k ∈ {8,…,15} they reported numerical evidence and wrote that a rigorous analytical proof was not available. We settle the first open case. Let φ₈ be the root of 72t³ - 189t² + 28t - 1 in (0,(1)/(10)). We prove that every feasible fully symmetric cell average decomposition for ℙ⁸ at θ = 0 has boundary weight at most φ₈, and we exhibit one that attains it, with four internal orbits and all data explicit in the real field obtained from ℚ(φ₈) by adjoining three square roots. Hence varpiₛₜₐᵣ(0,ℙ⁸) = varpiₛₜₐᵣ(0,ℙ⁹) = φ₈ and both conjectures hold at these two spaces. The rendered text of Cui–Ding–Wu records the value only as a decimal produced by an iterative solver; here it is an algebraic number of degree 3, and the decomposition attaining it is exact. The bound, the decomposition and the resulting value have been checked by the Lean 4 kernel; the step from them to the two conjectures is Cui–Ding–Wu's optimality criterion, quoted and not reproved.

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