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MPRK22(1) preserves positivity and total mass on positive inputs

Statement

For every finite conservative PDS Finite conservative production destruction systems with domain $Ω$ , every strictly positive input $y^{n} \in Ω_{+}$ , and every step size $h > 0$ , compute the Patankar Euler stage $z$ . If $z \in Ω_{+}$ , the stage and output of The positive-interior MPRK22(1) step are uniquely defined, strictly positive, and conserve total mass:

$z > 0, y^{n + 1} > 0, \sum_{i} z_{i} = \sum_{i} y_{i}^{n} = \sum_{i} y_{i}^{n + 1} .$

If $z \notin Ω_{+}$ , the algorithm returns rate-domain-failure before its second rate evaluation and makes no output claim. These preservation properties do not use the smoothness or compact-tube assumptions needed for the separate second-order error theorem.

Facts & Assumptions

Given: A finite conservative PDS with domain $Ω$ , $y^{n} \in Ω_{+}$ with $y^{n} > 0$ , $h > 0$ , and its Patankar Euler stage $z \in Ω_{+}$ .

[F1]

Finite conservative production destruction systems: $p_{i j} = d_{j i} \geq 0$ and the PDS uses only paired internal transfers.

[F2]

The positive-interior MPRK22(1) step: the stage solves $z = y^{n} + h A (y^{n}) z$ with $A (y^{n})$ having off-diagonal entries $p_{i j} (y^{n}) / y_{j}^{n}$ and diagonal entries that make each column sum zero; only if $z \in Ω_{+}$ does the algorithm evaluate $p_{i j} (z)$ in the final matrix $(I - h B (y^{n}, z))$ , where $B_{i j} = (p_{i j} (y^{n}) + p_{i j} (z)) / (2 z_{j})$ off the diagonal and each column of $B$ sums to zero.

[F3]

Strict column dominance gives an entrywise nonnegative inverse: a matrix with positive diagonal, nonpositive off-diagonal entries, and strict column diagonal dominance is invertible with nonnegative inverse; its inverse maps a positive vector to a positive vector.

Proof

1.1F1F2F3algebra

For the stage, $M^{(1)} : = I - h A (y^{n})$ has off-diagonal entries $- h p_{i j} (y^{n}) / y_{j}^{n} \leq 0$ and diagonal entries $1 + h \sum_{i \neq j} p_{i j} (y^{n}) / y_{j}^{n} > 0$ in column $j$ . The sum of the absolute off-diagonal entries in that column is $h \sum_{i \neq j} p_{i j} (y^{n}) / y_{j}^{n}$ , so its strict column-dominance margin is $1$ . By [F3], $z = (M^{(1)})^{- 1} y^{n} > 0$ . Also $\sum_{i} M_{i j}^{(1)} = 1$ for every column, hence $\sum_{i} z_{i} = \sum_{i} y_{i}^{n}$ .

2.1F2F3step 1.1algebra

Since $z > 0$ , every denominator in $B (y^{n}, z)$ is positive. Its off-diagonal entries are nonnegative, its diagonal entries are nonpositive, and its columns sum to zero by [F2]. Thus $M^{(2)} : = I - h B (y^{n}, z)$ has positive diagonal, nonpositive off-diagonal entries, and column diagonal-dominance margin $1$ . Applying [F3] to $M^{(2)}$ gives a unique $y^{n + 1} = (M^{(2)})^{- 1} y^{n} > 0$ .

3.1F2step 1.1step 2.1algebra∎

Every column of $M^{(2)}$ sums to one, so $\sum_{i} y_{i}^{n + 1} = \sum_{i} (M^{(2)} y^{n + 1})_{i} = \sum_{i} y_{i}^{n}$ . Together with step 1.1, this proves all stated positivity and conservation equalities.

Depends on

Used by

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

Cited to discharge well-definedness by The positive-interior MPRK22(1) step.

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Sources

Stefan Kopecz and Andreas Meister, On order conditions for modified Patankar-Runge-Kutta schemes, Lemmas 2.6–2.7 (standard reference, not scraped)
Sergio Blanes, Arieh Iserles, and Shev Macnamara, Positivity-preserving methods for ordinary differential equations, §3.4 (standard reference, not scraped)