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SSPRK(3,3) is third order and preserves conditional convex bounds

Statement

Assume the smoothness and compact-tube hypotheses of Sufficient explicit Runge–Kutta order conditions through order four. The step in The three-stage third-order SSP Runge–Kutta step has uniform local order at least three.

For an autonomous field, take an initial state $y \in C$ , where $C$ is a convex subset of the state domain, and let $E_{h} (u) = u + h f (u)$ be the forward-Euler map. If $E_{h} (C) \subseteq C$ and all displayed stages are admitted, then every SSPRK(3,3) stage and the output lie in $C$ . If $Φ : C \to R$ is convex, meaning $Φ (θ u + (1 - θ) v) \leq θ Φ (u) + (1 - θ) Φ (v)$ for $u, v \in C$ and $0 \leq θ \leq 1$ , and $Φ (E_{h} (u)) \leq Φ (u)$ for every $u \in C$ , then $Φ$ does not increase at any of the three stages or at the output.

For a nonautonomous field, assume the initial state $y \in C$ and that the forward-Euler maps at the actual stage start times $t$ , $t + h$ , and $t + h / 2$ preserve $C$ and satisfy the stated $Φ$ inequality on all admitted stage states. These conditions are premises of the theorem; the method alone does not assert positivity, total-variation diminishing, or any universal invariant-region property.

Facts & Assumptions

Given: The SSPRK(3,3) recurrence, an initial state $y \in C$ , the convex state set, the exact forward-Euler preservation hypotheses, and the smoothness/tube assumptions for the order claim.

[F1]

The three-stage third-order SSP Runge–Kutta step: $Y_{1} = E_{t, h} (y)$ , $Y_{2} = \frac{3}{4} y + \frac{1}{4} E_{t + h, h} (Y_{1})$ , and $y^{+} = \frac{1}{3} y + \frac{2}{3} E_{t + h / 2, h} (Y_{2})$ , where $E_{τ, h} (u) = u + h f (τ, u)$ ; in the autonomous case $E_{τ, h} = E_{h}$ .

[F2]

A convex subset of $R^{m}$ contains every line segment between two of its points: a convex set contains $θ u + (1 - θ) v$ whenever it contains $u, v$ and $0 \leq θ \leq 1$ .

[F3]

Explicit Runge–Kutta tableaus, stage maps, and failure status: the tableau has row-sum abscissae and the method output is the weighted sum of its stage slopes.

[F4]

Sufficient explicit Runge–Kutta order conditions through order four: the first four explicit tableau order equations through order three imply uniform local order at least three under the stated tube hypotheses.

[A1]

$C$ is convex, and the stated functional $Φ$ is convex on $C$ .

[A2]

The relevant forward-Euler maps send every admitted point of $C$ into $C$ and do not increase $Φ$ .

Proof

1.1F1F3algebra

Rewriting the recurrences in [F1] as explicit RK stages gives rows of $A$ equal to $(0, 0, 0)$ , $(1, 0, 0)$ , and $(1 / 4, 1 / 4, 0)$ , with $c = (0, 1, 1 / 2)$ and $b = (1 / 6, 1 / 6, 2 / 3)$ . Thus $\sum b_{i} = 1$ , $\sum b_{i} c_{i} = 1 / 2$ , $\sum b_{i} c_{i}^{2} = 1 / 3$ , and $\sum b_{i} (A c)_{i} = 1 / 6$ .

1.2givenA1A2F1F2construct

In the autonomous case $Y_{1} = E_{h} (y) \in C$ . Since $Y_{1} \in C$ , [A2] gives $E_{h} (Y_{1}) \in C$ ; [A1] and [F2] then give $Y_{2} = \frac{3}{4} y + \frac{1}{4} E_{h} (Y_{1}) \in C$ . Applying [A2] to $Y_{2}$ gives $E_{h} (Y_{2}) \in C$ , and [A1] gives $y^{+} = \frac{1}{3} y + \frac{2}{3} E_{h} (Y_{2}) \in C$ .

2.1F4step 1.1

The order equations in step 1.1 are exactly the order-one through order-three hypotheses of [F4], so the stated tube assumptions imply uniform local order at least three.

2.2givenA1A2F1step 1.2algebra

Convexity in [A1] gives $Φ (Y_{1}) \leq Φ (y)$ , $Φ (Y_{2}) \leq \frac{3}{4} Φ (y) + \frac{1}{4} Φ (E_{h} (Y_{1})) \leq Φ (y)$ , and $Φ (y^{+}) \leq \frac{1}{3} Φ (y) + \frac{2}{3} Φ (E_{h} (Y_{2})) \leq Φ (y)$ . Each last inequality uses [A2] at the relevant stage.

3.1givenA1A2F1F2step 1.2step 2.2∎

For a nonautonomous field the three formulas in [F1] use exactly $E_{t, h}$ at $y$ , $E_{t + h, h}$ at $Y_{1}$ , and $E_{t + h / 2, h}$ at $Y_{2}$ . Applying [A2] at those three start times and repeating the convex-combination calculations in steps 1.2–2.2 proves invariance and nonincrease.

Depends on

Used by

SSPRK(3,3) preserves the degree-zero upwind entropy bound Theorem

Dependency tree · two levels

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Sources

Sigal Gottlieb, Chi-Wang Shu, and Eitan Tadmor, Strong Stability-Preserving High-Order Time Discretization Methods, SIAM Review 43(1) (2001), §§2.1–2.2 and §4.1 (standard reference, not scraped)
Ernst Hairer, course notes, Chapter III: Équations Différentielles Ordinaires, §III.2 (standard reference, not scraped)