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MPRK22(1) has global second-order error on a compact positive tube

Statement

Let $T > 0$ , $N \geq 1$ , $h = T / N$ , and $t_{n} = n h$ . Use sets $U, V$ satisfying the domain and regularity hypotheses of MPRK22(1) has a uniform second-order defect and one-step bound on positive tubes, and let $K \subset U$ be a compact positive tube. Let $y : [0, T] \to U$ solve the finite conservative PDS Finite conservative production destruction systems, set $y_{0} = y (0)$ , and compute each step with The positive-interior MPRK22(1) step. Suppose the exact solution, all numerical states, and every MPRK22(1) stage remain in the common compact tube $K$ , and $h \leq h_{0}$ for the local constants in that lemma. Then a constant $C_{T}$ , independent of $N$ , satisfies

$\max_{0 \leq n \leq N} ∥ y_{n} - y (t_{n}) ∥_{2} \leq C_{T} h^{2} .$

This is a fixed-time, compact-tube convergence result; it does not assert a uniform error constant as the tube approaches a zero denominator or as stiffness parameters vary.

Facts & Assumptions

Given: A solution of the finite PDS on $[0, T]$ , exact initial data, the uniform grid, and the common-tube and step-size hypotheses above.

[F1]

Finite conservative production destruction systems: the ODE has the paired internal transfer form used by the methods.

[F2]

First-order systems, initial value problems, and solutions on intervals: the exact trajectory solves the stated first-order IVP.

[F3]

The positive-interior MPRK22(1) step: every step uses the Patankar Euler stage followed by the averaged-rate linear solve.

[F4]

MPRK22(1) preserves positivity and total mass on positive inputs: positive inputs whose stage lies in $Ω_{+}$ give uniquely defined positive stages and outputs for every $h > 0$ .

[F5]

MPRK22(1) has a uniform second-order defect and one-step bound on positive tubes: on the stated smooth positive tube and for $h \leq h_{0}$ , the exact-start defect is bounded by $C h^{3}$ and the step map has Lipschitz factor $1 + L h$ .

[F6]

Global error from uniform one-step defect and stability: for a fixed uniform grid, exact initial data, a common compact tube containing all states and stages, a uniform exact-start defect $C h^{p + 1}$ , and one-step factor $1 + L h$ , the global error is bounded by its stated exponential recurrence estimate of order $h^{p}$ .

Proof

1.1givenF1F2F3F4F5F6

Since $K \subset U \subset Ω \cap (0, \infty)^{m}$ and all iterates and stages remain in $K$ , every rate evaluation in [F3], including the evaluation at each stage, is in $Ω_{+}$ ; [F4] gives exact solvability and positivity independently of the order estimate. By [F5], the exact-start defect is uniformly at most $C h^{3}$ and the one-step stability factor is $1 + L h$ . The exact initial error is zero, and the common-tube assumption is exactly the remaining tube hypothesis of [F6].

2.1F5F6step 1.1algebra∎

Apply [F6] with $p = 2$ and $C_{0} = 0$ . Its $L > 0$ case gives $C (\exp (L T) - 1) h^{2} / L$ and its $L = 0$ case gives $C T h^{2}$ . In either case the coefficient is independent of $N$ , proving the asserted global $O (h^{2})$ error.

Depends on

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Sources

Stefan Kopecz and Andreas Meister, On order conditions for modified Patankar-Runge-Kutta schemes, §3.2 (standard reference, not scraped)
Ernst Hairer, course notes, Chapter III: Équations Différentielles Ordinaires (standard reference, not scraped)