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✓ 9 results · all verified · 5 also independently AI-judged

Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 4 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.

Positive Conservative Production Destruction Integrators

1 · Prerequisites

2 · Summary

Summary

For a finite closed system of nonnegative compartments, each production rate is paired with the equal destruction rate of its donor. The modified Patankar Euler step freezes these transfers and weights production and destruction by positive destination and source denominators, with rates evaluated at admissible states in the PDS domain. A finite matrix argument proves unique solvability, nonnegativity and exact total-mass conservation for any positive denominators. First-order accuracy is proved only for the standard choice $σ_{i} = y_{i}^{n}$ on a smooth compact positive tube; a conditional global first-order bound follows from the local defect and one-step stability estimate.

The usual ratio is undefined when a source compartment is zero. A separate source-factored, ratio-free step extends the first-order method to such states and proves its admissibility and local consistency, without assigning $0 / 0$ a value. The second-order item is specifically MPRK22(1): positivity and conservation hold for positive inputs at every step size when the stage rate evaluation is in the PDS domain; otherwise the procedure returns a rate-domain failure before that evaluation. Order two and global second-order error require a compact smooth positive tube with uniformly positive stage denominators. No order claim is made for zero-stage variants or for Patankar methods in general. The inexact-solve ledger makes positivity conditional on a certified component margin; a small residual alone does not establish it.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6-luna)Open item page →

Finite conservative production destruction systems

Definition

Fix an integer $m \geq 1$ and a domain $Ω \subseteq R^{m}$ ; its admissible nonnegative states are $Ω_{+} : = Ω \cap [0, \infty)^{m}$ . For every ordered pair $i, j \in {1, \dots, m}$ let $p_{i j}, d_{i j} : Ω \to R$ be production and destruction transfer rates that are nonnegative on $Ω_{+}$ . A finite conservative production destruction system (finite conservative PDS) is the autonomous ODE

$y_{i}^{'} = F_{i} (y) : = \sum_{j = 1}^{m} (p_{i j} (y) - d_{i j} (y)), i = 1, \dots, m,$

with internal-transfer identities

$p_{i j} (y) = d_{j i} (y), p_{i i} (y) = d_{i i} (y) = 0.$

The index $j$ is the donor/source of $p_{i j}$ and the index $i$ is its recipient/destination; $d_{j i}$ is the same transfer viewed as destruction from $j$ . The total mass is $M (y) : = \sum_{i = 1}^{m} y_{i}$ . The ODE is considered at states in $Ω_{+}$ . The definition describes a closed system: external sources and sinks must be specified separately and are not included in the conservation claims on this page. No existence, positivity, or regularity assertion about solutions is part of this definition.

LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-6-luna)Open item page →

Strict column dominance gives an entrywise nonnegative inverse

Statement

Let $M = (m_{i j}) \in R^{m \times m}$ for a finite $m \geq 1$ . Suppose $m_{j j} > 0$ , $m_{i j} \leq 0$ for $i \neq j$ , and

$m_{j j} > \sum_{i \neq j} ∣ m_{i j} ∣ (j = 1, \dots, m) .$

Then $M$ is invertible and every entry of $M^{- 1}$ is nonnegative. If $b \in R^{m}$ has nonnegative coordinates, then $M^{- 1} b \geq 0$ ; if every coordinate of $b$ is positive, then every coordinate of $M^{- 1} b$ is positive.

Facts & Assumptions

Given: A finite real matrix satisfying the displayed positive-diagonal, nonpositive-off-diagonal, and strict column-dominance hypotheses.

[F1]

The hypotheses give $c_{i j} : = - m_{i j} / m_{j j} \geq 0$ for $i \neq j$ and $c_{j j} = 0$ .

Proof

1.1givenF1algebraconstruct

Let $D = diag (m_{11}, \dots, m_{m m})$ and $C = I - M D^{- 1}$ . Then $D$ is invertible, $C \geq 0$ , and for each column $j$ , $r_{j} : = \sum_{i} C_{i j} = \sum_{i \neq j} ∣ m_{i j} ∣ / m_{j j} < 1$ . Since there are finitely many columns, $r : = \max_{j} r_{j} < 1$ .

2.1step 1.1algebrainduction

For every $k \geq 0$ , $C^{k} \geq 0$ and $\sum_{i} (C^{k})_{i j} \leq r^{k}$ for every column $j$ : it holds at $k = 0$ , and multiplying by $C$ gives $\sum_{i} (C^{k + 1})_{i j} = \sum_{ℓ} (\sum_{i} C_{i ℓ}) (C^{k})_{ℓ j} \leq r \sum_{ℓ} (C^{k})_{ℓ j}$ . Thus each entry series in $S : = \sum_{k = 0}^{\infty} C^{k}$ converges and is nonnegative.

3.1step 1.1step 2.1algebra

The finite geometric identity $(I - C) \sum_{k = 0}^{N} C^{k} = \sum_{k = 0}^{N} C^{k} (I - C) = I - C^{N + 1}$ holds by cancellation of adjacent powers. Step 2.1 gives $C^{N + 1} \to 0$ entrywise, so $(I - C) S = S (I - C) = I$ . Since $M = (I - C) D$ , $M^{- 1} = D^{- 1} S \geq 0$ , proving invertibility and inverse nonnegativity.

4.1step 3.1algebra∎

If $b \geq 0$ , then $M^{- 1} b \geq 0$ by step 3.1. If $b > 0$ , then $S \geq I$ entrywise, so $D^{- 1} S b \geq D^{- 1} b > 0$ coordinatewise. This proves both vector conclusions.

AlgorithmAlgorithm: Literature-sourcedProof: Not applicablejudge pass (gpt-6-luna)Open item page →

Modified Patankar Euler for conservative transfer systems

Definition

Let $m \geq 1$ , let $(p_{i j}, d_{i j})$ be the rates of a finite conservative PDS Finite conservative production destruction systems with state domain $Ω$ , and take an input state $y^{n} \in R^{m}$ , a step size $h$ , and declared denominators $σ_{i}$ . If $y^{n} \notin Ω_{+}$ , $h \leq 0$ , or any $σ_{i} \leq 0$ , return an inadmissible-input status before evaluating any rate. Otherwise the modified Patankar Euler step returns the unique solution $y^{n + 1}$ of

$y_{i}^{n + 1} = y_{i}^{n} + h \sum_{j \neq i} (p_{i j} (y^{n}) \frac{y_{j}^{n + 1}}{σ_{j}} - d_{i j} (y^{n}) \frac{y_{i}^{n + 1}}{σ_{i}}), i = 1, \dots, m .$

Equivalently, form $M \in R^{m \times m}$ by

$M_{i i} = 1 + h \sum_{j \neq i} \frac{d_{i j} (y^{n})}{σ_{i}}, M_{i j} = - h \frac{p_{i j} (y^{n})}{σ_{j}} (i \neq j),$

and solve $M y^{n + 1} = y^{n}$ . The rates in the solve are evaluated only at the admissible state $y^{n} \in Ω_{+}$ . For every input passing the branch, return the unique exact solution of the stated linear system. The standard first-order choice is $σ_{i} = y_{i}^{n}$ , which is admissible only when every $y_{i}^{n} > 0$ . A separately defined rate-factor form extends a first-order step to zero donor coordinates under an additional source-factorization hypothesis.

DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-6-luna)Open item page →

Source-factored transfer rates for zero compartments

Definition

Let $(p_{i j}, d_{i j})$ be the rates of a finite conservative PDS Finite conservative production destruction systems. A source-factored transfer representation on an open set $U \subseteq Ω \subseteq R^{m}$ containing a compact set $K \subseteq [0, \infty)^{m}$ consists of functions $a_{i j} : U \to R$ , $i \neq j$ , such that each $a_{i j}$ is $C^{2}$ on the open subset $U \subseteq R^{m}$ in the multivariable sense of $C^{k}$ maps and multi-index derivative notation in Euclidean space, is nonnegative on $U \cap [0, \infty)^{m}$ , and, for every $y \in U \cap [0, \infty)^{m}$ ,

$p_{i j} (y) = y_{j} a_{i j} (y), d_{i j} (y) = y_{i} a_{j i} (y), a_{i i} : = 0.$

Here $a_{i j}$ is the transfer rate from donor $j$ to recipient $i$ per unit of donor quantity. Nonnegativity is required on admissible nonnegative states; the $C^{2}$ extension to the open neighborhood is used for local analysis and need not be nonnegative at inadmissible states with negative coordinates. The factorization is an additional hypothesis, not a consequence of nonnegative rates and the identity $p_{i j} = d_{j i}$ . On strictly positive states the ratio $p_{i j} (y) / y_{j}$ equals $a_{i j} (y)$ ; at a zero donor coordinate the factorized coefficient remains defined and avoids the indeterminate expression $0 / 0$ .

AlgorithmAlgorithm: AI-adaptedProof: Not applicablejudge pass (gpt-6-luna)Open item page →

Zero-compatible modified Patankar Euler in rate-factor form

Definition

Let $K$ be a compact nonnegative state set with a source-factored transfer representation Source-factored transfer rates for zero compartments. Given $y^{n} \in R^{m}$ and a step size $h$ , if $y^{n} \notin K$ or $h \leq 0$ , return an inadmissible-input status before evaluating any factor. Otherwise form $B (y^{n}) \in R^{m \times m}$ by

$B_{i j} (y^{n}) = a_{i j} (y^{n}) (i \neq j), B_{j j} (y^{n}) = - \sum_{i \neq j} a_{i j} (y^{n}) .$

Return the unique solution $y^{n + 1}$ of

$(I - h B (y^{n})) y^{n + 1} = y^{n} .$

Equivalently, component $i$ satisfies

$y_{i}^{n + 1} = y_{i}^{n} + h \sum_{j \neq i} (a_{i j} (y^{n}) y_{j}^{n + 1} - a_{j i} (y^{n}) y_{i}^{n + 1}) .$

This procedure uses no denominator $y_{j}^{n}$ and is defined when some donor coordinates vanish. It is the factorized extension of the first-order Patankar Euler formula to the zero-state boundary. In exact arithmetic its linear system is uniquely solvable by Strict column dominance gives an entrywise nonnegative inverse.

AlgorithmAlgorithm: Literature-sourcedProof: Not applicableOpen item page →

The positive-interior MPRK22(1) step

Definition

Let $(p_{i j}, d_{i j})$ be the rates of a finite conservative PDS Finite conservative production destruction systems with state domain $Ω$ . Given $y^{n} \in R^{m}$ and $h$ , if $y^{n} \notin Ω_{+}$ , any coordinate of $y^{n}$ is nonpositive, or $h \leq 0$ , return an inadmissible-input status before evaluating rates or solving a system. First compute the stage $z$ by invoking Modified Patankar Euler for conservative transfer systems with denominators $σ_{i} = y_{i}^{n}$ :

$z_{i} = y_{i}^{n} + h \sum_{j \neq i} (p_{i j} (y^{n}) \frac{z_{j}}{y_{j}^{n}} - d_{i j} (y^{n}) \frac{z_{i}}{y_{i}^{n}}) .$

If the stage solve does not return a unique vector, return stage-failure. If its output $z \notin Ω_{+}$ , return rate-domain-failure before evaluating rates at $z$ . Otherwise form $B (y^{n}, z)$ with, for $i \neq j$ ,

$B_{i j} (y^{n}, z) : = \frac{p_{i j} (y^{n}) + p_{i j} (z)}{2 z_{j}}, B_{j j} (y^{n}, z) : = - \sum_{i \neq j} B_{i j} (y^{n}, z),$

and solve

$(I - h B (y^{n}, z)) y^{n + 1} = y^{n} .$

Equivalently,

$y_{i}^{n + 1} = y_{i}^{n} + \frac{h}{2} \sum_{j \neq i} ((p_{i j} (y^{n}) + p_{i j} (z)) \frac{y_{j}^{n + 1}}{z_{j}} - (d_{i j} (y^{n}) + d_{i j} (z)) \frac{y_{i}^{n + 1}}{z_{i}}) .$

If the final solve is not uniquely solvable, return update-failure; otherwise return its unique solution $y^{n + 1}$ . No zero-stage or denominator-regularized second-order variant is defined here.

TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-6-luna)Open item page →

Modified Patankar Euler preserves positivity and total mass

Statement

For a finite conservative PDS Finite conservative production destruction systems with domain $Ω$ , an admissible input $y^{n} \in Ω_{+}$ , step $h > 0$ , and any positive denominators $σ_{i} > 0$ , the step of Modified Patankar Euler for conservative transfer systems is uniquely defined, satisfies $y^{n + 1} \geq 0$ , and conserves total mass:

$\sum_{i = 1}^{m} y_{i}^{n + 1} = \sum_{i = 1}^{m} y_{i}^{n} .$

If $y^{n} > 0$ coordinatewise, then $y^{n + 1} > 0$ coordinatewise. These conclusions do not assert order for arbitrary denominators; the first-order choice and its accuracy hypotheses are treated separately.

Facts & Assumptions

Given: The finite conservative PDS identities, $y^{n} \in Ω_{+}$ , $h > 0$ , and $σ_{i} > 0$ for every compartment.

[F1]

Finite conservative production destruction systems: $p_{i j} = d_{j i} \geq 0$ , $p_{i i} = d_{i i} = 0$ , and $y_{i}^{'} = \sum_{j} (p_{i j} - d_{i j})$ .

[F2]

Modified Patankar Euler for conservative transfer systems: $M y^{n + 1} = y^{n}$ , with $M_{i i} = 1 + h \sum_{k \neq i} d_{i k} (y^{n}) / σ_{i}$ and $M_{i j} = - h p_{i j} (y^{n}) / σ_{j}$ for $i \neq j$ .

[F3]

Strict column dominance gives an entrywise nonnegative inverse: a finite matrix with positive diagonal, nonpositive off-diagonal entries, and strict column diagonal dominance has a nonnegative inverse; the inverse maps positive vectors to positive vectors.

Proof

1.1F1F2F3algebra

For every $j$ , $M_{j j} > 0$ and $M_{i j} \leq 0$ when $i \neq j$ by [F1]–[F2]. Moreover, $M_{j j} - \sum_{i \neq j} ∣ M_{i j} ∣ = 1 + \frac{h}{σ_{j}} \sum_{k \neq j} d_{j k} (y^{n}) - \frac{h}{σ_{j}} \sum_{i \neq j} p_{i j} (y^{n}) = 1$ , because $p_{i j} = d_{j i}$ makes the two finite sums equal. Thus [F3] applies: $M$ is invertible and $y^{n + 1} = M^{- 1} y^{n} \geq 0$ , with strict positivity when $y^{n} > 0$ .

2.1F1F2step 1.1algebra∎

The same identity gives each column sum $\sum_{i} M_{i j} = 1$ . Therefore $\sum_{i} y_{i}^{n} = \sum_{i} (M y^{n + 1})_{i} = \sum_{j} (\sum_{i} M_{i j}) y_{j}^{n + 1} = \sum_{j} y_{j}^{n + 1}$ . This proves exact summed conservation.

LemmaStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-6-luna)Open item page →

Modified Patankar Euler has a uniform first-order defect and one-step bound on positive tubes

Statement

Let $F$ be the vector field of a finite conservative PDS Finite conservative production destruction systems on a bounded open convex set $U \subset Ω \cap R_{> 0}^{m}$ with $\overline{U} \subset R_{> 0}^{m}$ . Suppose there is an open set $V$ with $\overline{U} \subset V \subset Ω \cap R_{> 0}^{m}$ on which the transfer rates are multivariable $C^{2}$ and their derivatives through order two are bounded. For $y \in U$ , let $Φ_{h} (y)$ be Modified Patankar Euler for conservative transfer systems with the standard denominators $σ_{i} = y_{i}$ . Then there are constants $C, L, h_{0} > 0$ such that, for $0 < h \leq h_{0}$ :

for every exact solution $y (t)$ whose segment and the method states under consideration remain in a compact subset of $U$ , the exact-start local defect satisfies $∥ Φ_{h} (y (t)) - y (t + h) ∥_{2} \leq C h^{2}$ ; and
for all $u, v \in U$ , $∥ Φ_{h} (u) - Φ_{h} (v) ∥_{2} \leq (1 + L h) ∥ u - v ∥_{2}$ .

The defect is the method output minus the exact next state and the second inequality is one-step Lipschitz stability in the convention of One-step ODE methods, local defect, uniform order, and step stability. This result is restricted to a compact positive tube; it does not define the standard ratio formula at zero compartments.

Facts & Assumptions

Given: A finite conservative PDS, the stated multivariable $C^{2}$ rate data on $V \subset Ω \cap R_{> 0}^{m}$ , an exact solution segment and method states in a compact subset of $U$ , and $h > 0$ sufficiently small.

[F1]

Finite conservative production destruction systems: $F_{i} (y) = \sum_{j} (p_{i j} (y) - d_{i j} (y))$ , $p_{i j} = d_{j i} \geq 0$ , and $p_{i i} = d_{i i} = 0$ .

[F2]

Modified Patankar Euler for conservative transfer systems with $σ_{i} = y_{i}$ is the linear system $Φ_{h} (y) = (I - h A (y))^{- 1} y$ , where $A_{i j} (y) = p_{i j} (y) / y_{j}$ for $i \neq j$ and $A_{j j} (y) = - \sum_{i \neq j} p_{i j} (y) / y_{j}$ .

[F3]

Strict column dominance gives an entrywise nonnegative inverse: the Patankar matrix $I - h A (y)$ is invertible; its inverse is entrywise nonnegative.

[F4]

One-step ODE methods, local defect, uniform order, and step stability: local defect is the exact-start method output minus the exact next state; uniform local order one means a uniform $O (h^{2})$ bound; one-step stability is the stated $(1 + L h)$ Lipschitz bound.

[F5]

$C^{k}$ maps and multi-index derivative notation in Euclidean space: $C^{2}$ means that coordinate derivatives through order two exist and are continuous.

[F6]

If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative: continuous first coordinate partials on an open set give the total derivative with those partials as its Jacobian; applying it to each first partial also supplies continuous total second derivatives for a $C^{2}$ map.

[F7]

The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ and Chain sum product and composition rules for Banach derivatives: total derivatives obey the chain rule, and coordinate products obey the bounded-bilinear product rule.

[F8]

The matrix norm induced by a published vector p-norm at $p = 2$ : $∥ A ∥_{2} = \sup_{∥ w ∥_{2} \leq 1} ∥ A w ∥_{2}$ is the Euclidean induced operator norm and is a matrix norm.

[F9]

Induced matrix norms are compatible with matrix-vector multiplication, submultiplicative, and satisfy ||I|| = 1: $∥ I ∥_{2} = 1$ and the induced norm is compatible with matrix products.

[F10]

On a convex open set, a uniform bound $∥ D f (z) v ∥_{2} \leq M ∥ v ∥_{2}$ implies $∥ f (y) - f (x) ∥_{2} \leq M ∥ y - x ∥_{2}$ : on an open convex set, a uniform derivative operator bound $∥ D g (z) w ∥_{2} \leq M ∥ w ∥_{2}$ implies $∥ g (u) - g (v) ∥_{2} \leq M ∥ u - v ∥_{2}$ .

[F11]

Multivariable Taylor formula with a Lagrange remainder along a line segment: a $C^{2}$ function on an open convex set has a first-order Taylor expansion with a remainder bounded uniformly by its second-derivative bound on the intervening segment.

Proof

1.1givenF1F2F5F6F7algebra

Since $U \subset R_{> 0}^{m}$ , each coordinate $y_{j}$ stays bounded away from zero on $\overline{U}$ . On $R ∖ {0}$ the reciprocal $s \mapsto s^{- 1}$ has derivatives $- s^{- 2}$ and $2 s^{- 3}$ , obtained from the difference quotients $- 1 / (s (s + t))$ and $(2 s + t) / (s^{2} (s + t)^{2})$ ; composing with the coordinate projection gives continuous first and second partials for $y \mapsto y_{j}^{- 1}$ . Applying the product rule [F7] twice shows every entry $A_{i j} (y) = p_{i j} (y) y_{j}^{- 1}$ ( $i \neq j$ ), and hence every diagonal sum, is $C^{2}$ on $U$ ; [F6] applied to these entries and their first partials supplies continuous total derivatives through order two. The row equations give $A (y) y = F (y)$ : the off-diagonal terms contribute $\sum_{j \neq i} p_{i j}$ , and the diagonal term is $- \sum_{j \neq i} d_{i j}$ . For column $j$ , $\sum_{i} A_{i j} = 0$ because $A_{j j} = - \sum_{i \neq j} A_{i j}$ .

2.1givenF1F3F5F6F7algebra

The off-diagonal entries of $A$ are nonnegative and its diagonal entries are nonpositive. Thus $I - h A (y)$ has positive diagonal, nonpositive off-diagonal entries, and strict column diagonal-dominance margin $1$ ; [F3] makes it invertible. On $[0, h_{0}] \times \overline{U}$ , the entries of $I - h A (y)$ and its determinant are $C^{2}$ ; the determinant is nonzero by [F3]. Its cofactors and determinant are finite sums of products of entries, and the scalar reciprocal of the determinant is $C^{2}$ by the same reciprocal calculation as in step 1.1. The adjugate formula therefore makes the inverse a $C^{2}$ matrix-valued function there; compactness gives uniform bounds for the inverse and its first derivatives.

3.1givenF1F2F4F5F6F7F11step 2.1algebra

The matrix identity $(I - h A)^{- 1} = I + h A + h^{2} A^{2} (I - h A)^{- 1}$ yields $Φ_{h} (y) = y + h F (y) + h^{2} G (h, y)$ with $G (h, y) : = A (y)^{2} (I - h A (y))^{- 1} y$ . Step 2.1 and compactness bound $G$ uniformly. Along the exact solution, [F5]–[F7] give $y^{''} (t) = D F (y (t)) F (y (t))$ , which is uniformly bounded on the compact tube. Taylor's formula [F11] gives $y (t + h) = y (t) + h F (y (t)) + O (h^{2})$ uniformly. Subtracting the expansions proves the local-defect estimate in [F4].

4.1givenF5F7F8F9step 2.1step 3.1algebra

The function $G$ in step 3.1 is continuously differentiable in $y$ with uniformly bounded derivative on $\overline{U}$ by [F5]–[F7] and step 2.1. Differentiating that identity gives $D_{y} Φ_{h} (y) = I + h D F (y) + h^{2} D_{y} G (h, y)$ . Compactness bounds $∥ D F (y) ∥_{2}$ by $L_{0}$ and $∥ D_{y} G (h, y) ∥_{2}$ by $G_{0}$ . The matrix-norm triangle inequality from [F8] and $∥ I ∥_{2} = 1$ from [F9] yield $∥ D_{y} Φ_{h} (y) ∥_{2} \leq 1 + h L_{0} + h^{2} G_{0} \leq 1 + (L_{0} + h_{0} G_{0}) h$ for $0 < h \leq h_{0}$ .

5.1F4F10step 4.1∎

Apply [F10] to $Φ_{h}$ on the convex open set $U$ using the uniform derivative bound from step 4.1. This gives $∥ Φ_{h} (u) - Φ_{h} (v) ∥_{2} \leq (1 + L h) ∥ u - v ∥_{2}$ for every $u, v \in U$ , exactly the one-step stability assertion of [F4].

TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-6-luna)Open item page →

The factorized zero-state step is well-defined, nonnegative, conservative, and locally first order

Statement

Let a finite conservative PDS have a source-factored representation on an open neighborhood $U \subseteq Ω$ of a compact set $K \subseteq [0, \infty)^{m}$ , with factors that are multivariable $C^{2}$ on $U$ and nonnegative on $U \cap [0, \infty)^{m}$ , as in Source-factored transfer rates for zero compartments. For every $y^{n} \in K$ and $h > 0$ , Zero-compatible modified Patankar Euler in rate-factor form has a unique output $y^{n + 1} \geq 0$ and conserves $\sum_{i} y_{i}$ . For the accuracy clause, fix a compact set $K_{*} \subset U$ containing $K$ and all exact solution segments under consideration, and define the $C^{2}$ extension ${\tilde{F}}_{i} (y) : = \sum_{j \neq i} (y_{j} a_{i j} (y) - y_{i} a_{j i} (y))$ on $U$ . If those exact segments remain in $K_{*} \cap [0, \infty)^{m}$ , then the exact-start local defect is uniformly $O (h^{2})$ as $h ↓ 0$ , including when some coordinates of $y^{n}$ are zero. The claim is local; no global error bound at the boundary is asserted.

Facts & Assumptions

Given: A finite conservative PDS, a source-factored representation on a neighborhood of $K$ , $y^{n} \in K \subseteq [0, \infty)^{m}$ , $h > 0$ , and the fixed compact tube $K_{*} \subset U$ from the defect clause when local accuracy is asserted.

[F1]

Finite conservative production destruction systems: $F_{i} (y) = \sum_{j \neq i} (p_{i j} (y) - d_{i j} (y))$ and the pair rates satisfy $p_{i j} = d_{j i}$ .

[F2]

Source-factored transfer rates for zero compartments: $p_{i j} (y) = y_{j} a_{i j} (y)$ and $d_{i j} (y) = y_{i} a_{j i} (y)$ with $C^{2}$ factors that are nonnegative on admissible nonnegative states, including at zero coordinates.

[F3]

Zero-compatible modified Patankar Euler in rate-factor form: the output solves $(I - h B (y^{n})) y^{n + 1} = y^{n}$ , where $B_{i j} = a_{i j}$ for $i \neq j$ and $B_{j j} = - \sum_{i \neq j} a_{i j}$ .

[F4]

Strict column dominance gives an entrywise nonnegative inverse: a strict column-diagonally-dominant matrix with positive diagonal and nonpositive off-diagonal entries has an entrywise nonnegative inverse.

[F5]

First-order systems, initial value problems, and solutions on intervals: an exact trajectory satisfies $y^{'} = F (y)$ on its interval.

[F6]

$C^{k}$ maps and multi-index derivative notation in Euclidean space and If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative: continuous first partials give a total derivative, and the $C^{2}$ factors give continuous first derivatives.

[F7]

Chain sum product and composition rules for Banach derivatives and The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ : products of differentiable coordinate maps obey the bounded-bilinear product rule, and composition obeys the total chain rule.

[F8]

Multivariable Taylor formula with a Lagrange remainder along a line segment: a $C^{2}$ solution curve on a time interval has a first-order Taylor expansion with remainder bounded by its second derivative on that interval.

[F9]

One-step ODE methods, local defect, uniform order, and step stability: the exact-start local defect is the method output minus the exact next state.

Proof

1.1givenF2F3F4algebra

For each column $j$ , $B_{j j} = - \sum_{i \neq j} a_{i j} \leq 0$ , $B_{i j} = a_{i j} \geq 0$ off the diagonal, and $\sum_{i} B_{i j} = 0$ . Therefore $M : = I - h B (y^{n})$ has $M_{j j} = 1 + h \sum_{i \neq j} a_{i j} (y^{n}) > 0$ , $M_{i j} = - h a_{i j} (y^{n}) \leq 0$ for $i \neq j$ , and $M_{j j} - \sum_{i \neq j} ∣ M_{i j} ∣ = 1$ . By [F4], $M^{- 1} \geq 0$ ; hence the unique solution $y^{n + 1} = M^{- 1} y^{n}$ exists and is nonnegative, including for zero coordinates in $y^{n}$ .

2.1F3step 1.1algebra

The column sums of $M$ equal one, so $\sum_{i} y_{i}^{n} = \sum_{i} (M y^{n + 1})_{i} = \sum_{j} (\sum_{i} M_{i j}) y_{j}^{n + 1} = \sum_{j} y_{j}^{n + 1}$ .

2.2F1F2F3algebra

The factorization [F2] gives $B (y) y = F (y)$ for every $y \in K$ : its $i$ -th component is $\sum_{j \neq i} a_{i j} (y) y_{j} - y_{i} \sum_{j \neq i} a_{j i} (y) = \sum_{j \neq i} (p_{i j} (y) - d_{i j} (y))$ . The matrix identity used in step 1.1 also gives $(I - h B (y))^{- 1} = I + h B (y) + h^{2} B (y)^{2} (I - h B (y))^{- 1}$ .

3.1givenF2F5F6F7F8F9step 2.2algebra∎

The factors are $C^{2}$ on $U$ , so the coordinate products $y_{j} a_{i j} (y)$ and $y_{i} a_{j i} (y)$ have continuous partial derivatives through order two by applying the bounded-bilinear product rule [F7] twice. Applying the continuous-partials theorem [F6] to these components and their first partials shows that $\tilde{F}$ is $C^{2}$ and has continuous first total derivative on $U$ . On the compact set $[0, h_{0}] \times K$ of admissible nonnegative starting states, $B$ is bounded and $I - h B (y^{n})$ is invertible by step 1.1 for $h > 0$ and equals $I$ at $h = 0$ . Its inverse is continuous there, hence uniformly bounded for sufficiently small $h$ . The matrix identity from step 2.2 then yields $y^{n + 1} = y^{n} + h F (y^{n}) + O (h^{2})$ uniformly, without dividing by any coordinate of $y^{n}$ . On the nonnegative exact path, the PDS field equals the $C^{2}$ extension $\tilde{F}$ defined in the Statement, so [F5]–[F7] give $y^{'} = \tilde{F} (y)$ and bounded $y^{''} = D \tilde{F} (y) \tilde{F} (y)$ on $K_{*}$ . Applying [F8] to the time curve gives the same expansion uniformly. Their difference is therefore $O (h^{2})$ uniformly, which is the local-defect bound specified by [F9].

TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passOpen item page →

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.

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MPRK22(1) has a uniform second-order defect and one-step bound on positive tubes

Statement

Let $F$ be the vector field of a finite conservative PDS Finite conservative production destruction systems on a bounded open convex set $U \subset Ω \cap R_{> 0}^{m}$ with $\overline{U} \subset R_{> 0}^{m}$ . Assume there is an open set $V$ with $\overline{U} \subset V \subset Ω \cap R_{> 0}^{m}$ on which the transfer rates are multivariable $C^{2}$ , with derivatives through order two bounded there. Let $Ψ_{h} (y)$ be The positive-interior MPRK22(1) step. For sufficiently small $h > 0$ , there are constants $C, L, h_{0} > 0$ such that:

for every exact solution segment in a compact subset of $U$ , the exact-start defect satisfies $∥ Ψ_{h} (y (t)) - y (t + h) ∥_{2} \leq C h^{3}$ ; and
for all $u, v \in U$ , $∥ Ψ_{h} (u) - Ψ_{h} (v) ∥_{2} \leq (1 + L h) ∥ u - v ∥_{2}$ .

The local order-two estimate requires the positive lower bound on the stage denominators and the stated smoothness. Positivity and conservation for every $h > 0$ are separate preservation properties and are not used to claim the local error order.

Facts & Assumptions

Given: A finite conservative PDS, the stated multivariable $C^{2}$ rate data on $V \subset Ω \cap R_{> 0}^{m}$ , an exact solution segment in the tube, and a sufficiently small step size.

[F1]

Finite conservative production destruction systems: $F_{i} = \sum_{j} (p_{i j} - d_{i j})$ , $p_{i j} = d_{j i} \geq 0$ , and $p_{i i} = d_{i i} = 0$ .

[F2]

Modified Patankar Euler for conservative transfer systems: with $σ_{i} = y_{i}$ , the first stage is $z = (I - h A (y))^{- 1} y$ , where $A (y) y = F (y)$ .

[F3]

The positive-interior MPRK22(1) step: the final matrix has off-diagonal coefficients $(p_{i j} (y) + p_{i j} (z)) / (2 z_{j})$ , zero column sums before subtracting from $I$ , and update $Ψ_{h} (y) = (I - h B (y, z))^{- 1} y$ .

[F4]

Strict column dominance gives an entrywise nonnegative inverse: each stage/final matrix with positive declared denominators is invertible with nonnegative inverse, and its inverse maps a strictly positive right-hand side to a strictly positive solution.

[F5]

First-order systems, initial value problems, and solutions on intervals: an exact trajectory satisfies $y^{'} = F (y)$ .

[F6]

$C^{k}$ maps and multi-index derivative notation in Euclidean space and If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative: continuous coordinate partials give total derivatives; applying the theorem to first partials also supplies the second total derivatives needed here.

[F7]

The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ and Chain sum product and composition rules for Banach derivatives: compositions and bounded-bilinear products obey their total-derivative rules, including when applied to derivatives.

[F8]

Multivariable Taylor formula with a Lagrange remainder along a line segment and The Lagrange and Cauchy forms of Taylor's remainder: the rate vector and exact solution have the Taylor expansions with uniform remainders on the compact tube.

[F9]

One-step ODE methods, local defect, uniform order, and step stability: exact-start defect and one-step stability have the conventions used in this statement.

[F10]

The matrix norm induced by a published vector p-norm at $p = 2$ and Induced matrix norms are compatible with matrix-vector multiplication, submultiplicative, and satisfy ||I|| = 1: the induced Euclidean operator norm is a matrix norm and $∥ I ∥_{2} = 1$ .

[F11]

On a convex open set, a uniform bound $∥ D f (z) v ∥_{2} \leq M ∥ v ∥_{2}$ implies $∥ f (y) - f (x) ∥_{2} \leq M ∥ y - x ∥_{2}$ : a uniform derivative bound on a convex open set implies the corresponding Lipschitz bound.

Proof

1.1givenF1F2F4F6F7algebra

Define $A (y)$ as in [F2]. On $U$ , the coordinate reciprocals $y_{j}^{- 1}$ have continuous partials $\partial_{k} (y_{j}^{- 1}) = - δ_{j k} y_{j}^{- 2}$ and $\partial_{ℓ} \partial_{k} (y_{j}^{- 1}) = 2 δ_{j k} δ_{ℓ j} y_{j}^{- 3}$ , as follows from the scalar reciprocal difference quotients on $R ∖ {0}$ ; the product rule [F7] then makes every entry of $A$ $C^{2}$ from the rate data [F6]. By the continuous-partials theorem [F6], these entries have continuous total derivatives through order two. Direct multiplication gives $A (y) y = F (y)$ . The matrix $I - h A (y)$ is invertible for $h > 0$ by [F4] and equals $I$ at $h = 0$ ; its determinant and cofactors are $C^{2}$ finite sums of products, and its determinant is nonzero, so the adjugate formula and the scalar reciprocal calculation make its inverse $C^{2}$ on the compact set of admitted $(h, y)$ . It is uniformly bounded there for small $h$ . Since $y > 0$ , [F4] gives $z = (I - h A (y))^{- 1} y > 0$ . The identity $(I - h A)^{- 1} = I + h A + h^{2} A^{2} (I - h A)^{- 1}$ now gives $z = y + h F (y) + O (h^{2})$ uniformly; in particular $z - y = O (h)$ , so after decreasing $h_{0}$ we have $z \in V \subset Ω$ uniformly for $y \in U$ . Thus all stage denominators and the final rate evaluations are valid.

2.1F1F2F3F4F6F7step 1.1algebra

Write $B = B (y, z)$ for the final coefficient matrix in [F3]. Its entries are $C^{2}$ on the compact region of pairs produced inside $V$ by step 1.1: the rates are $C^{2}$ , the stage map is $C^{2}$ , and each reciprocal $z_{j}^{- 1}$ has the same coordinate derivative formulas as in step 1.1. Its columns sum to zero, and direct multiplication using $p_{i j} = d_{j i}$ gives $B (y, z) z = \frac{1}{2} (F (y) + F (z))$ . For small $h$ , step 1.1 keeps every $z_{j}$ bounded away from zero, so $B$ and its first derivatives are uniformly bounded. Also $B (y, y) = A (y)$ ; the mean-value theorem on the compact tube and $z - y = O (h)$ give $B (y, z) = A (y) + O (h)$ . The final matrix $I - h B$ is invertible by [F4] and its inverse is uniformly bounded by the adjugate formula. Expanding its inverse yields $Ψ_{h} (y) = y + h B (y, z) y + O (h^{2}) = y + h F (y) + O (h^{2})$ . Comparing with step 1.1 gives $Ψ_{h} (y) - z = O (h^{2})$ .

2.2givenF1F2F3F4F6F7step 1.1algebra

The stage and final maps are $C^{1}$ in $y$ near the compact positive tube: their rates are $C^{2}$ , their denominators stay bounded away from zero, and their matrix inverses are $C^{2}$ by the adjugate formula and the reciprocal calculation in steps 1.1–2.1. For any direction $v$ , differentiating $z = y + h A (y) z$ gives $(I - h A) D_{y} z [v] = v + h (D A (y) [v]) z$ ; boundedness from step 1.1 therefore gives $D_{y} z [v] = v + O (h)$ uniformly for $∥ v ∥_{2} \leq 1$ .

3.1F1F5F6F7F8F9step 1.1step 2.1algebra

Using $Ψ_{h} - y = h B Ψ_{h}$ , substitute $z$ and use step 2.1 to obtain $Ψ_{h} - y = h B z + h B (Ψ_{h} - z) = \frac{h}{2} (F (y) + F (z)) + O (h^{3})$ . Taylor expansion of $F$ at $y$ , together with $z - y = h F (y) + O (h^{2})$ from step 1.1, gives $F (z) = F (y) + h D F (y) F (y) + O (h^{2})$ . Thus $Ψ_{h} (y) = y + h F (y) + \frac{h^{2}}{2} D F (y) F (y) + O (h^{3})$ . By [F5]–[F7], the exact solution has the same expansion: $y^{''} = D F (y) F (y)$ and its third derivative is bounded, so the one-variable Taylor remainder applied coordinatewise is $O (h^{3})$ . Their difference is the asserted uniform exact-start defect in [F9].

3.2F3F4F6F7step 2.1step 2.2algebra

Differentiating $Ψ_{h} = y + h B (y, z) Ψ_{h}$ in direction $v$ gives $D_{y} Ψ_{h} [v] = (I - h B (y, z))^{- 1} (v + h (D_{1} B (y, z) [v] + D_{2} B (y, z) [D_{y} z [v]]) Ψ_{h})$ , with the inverse defined by step 2.1.

4.1F1F2F3F6F7step 1.1step 2.1step 2.2step 3.2algebra

Steps 1.1–2.2 give $z = y + O (h)$ , $Ψ_{h} = y + O (h)$ , $B (y, z) = A (y) + O (h)$ , and $(I - h B)^{- 1} = I + h B + O (h^{2})$ uniformly. Substitution yields $D_{y} Ψ_{h} [v] = v + h (A (y) v + (D_{1} B (y, y) [v] + D_{2} B (y, y) [v]) y) + o (h)$ . Differentiating $B (y, y) y = F (y)$ with the chain and product rules identifies the coefficient as $D F (y) v$ ; compactness makes the remainder uniform, so $D_{y} Ψ_{h} (y) = I + h D F (y) + h E (h, y)$ with $\sup_{y \in U} ∥ E (h, y) ∥_{2} \to 0$ .

5.1F10F11step 4.1algebra∎

Compactness bounds $∥ D F (y) ∥_{2}$ uniformly, and step 4.1 makes the $E$ contribution at most $h$ for sufficiently small $h$ . The norm properties in [F10] then give $∥ D_{y} Ψ_{h} (y) ∥_{2} \leq 1 + L h$ after decreasing $h_{0}$ and increasing $L$ . Applying [F11] on convex $U$ proves $∥ Ψ_{h} (u) - Ψ_{h} (v) ∥_{2} \leq (1 + L h) ∥ u - v ∥_{2}$ .

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Modified Patankar Euler converges globally with first-order error on a 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 Modified Patankar Euler has a uniform first-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 ODE of a finite conservative PDS Finite conservative production destruction systems, let $y_{0} = y (0)$ , and obtain $y_{n + 1}$ by Modified Patankar Euler for conservative transfer systems with $σ_{i} = (y_{n})_{i}$ . Suppose the exact path and every numerical state lie in the common compact tube $K$ , and $h \leq h_{0}$ for the local constants in that lemma. Then there is a constant $C_{T}$ , independent of $N$ , such that

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

The conclusion is conditional on the stated positive compact tube; positivity and mass conservation alone do not establish that a trajectory remains in such a tube for a prescribed interval.

Facts & Assumptions

Given: A solution of the finite PDS on $[0, T]$ , exact initial data, a fixed uniform grid, and the assumption that exact and numerical states stay in the common compact positive tube $K$ .

[F1]

Finite conservative production destruction systems: the ODE has finite paired internal transfer rates.

[F2]

First-order systems, initial value problems, and solutions on intervals: the exact trajectory is a solution of the stated first-order IVP on its interval.

[F3]

Modified Patankar Euler for conservative transfer systems: the standard first-order step uses $σ_{i} = (y_{n})_{i}$ and solves its frozen-rate linear system.

[F4]

Modified Patankar Euler has a uniform first-order defect and one-step bound on positive tubes: on $K$ and for $h \leq h_{0}$ , the exact-start defect is at most $C h^{2}$ and the step map is Lipschitz with factor $1 + L h$ .

[F5]

Global error from uniform one-step defect and stability: on a fixed grid and a common compact tube containing the exact path and numerical states, initial error $C_{0} h^{p}$ , exact-start defect $C h^{p + 1}$ , and one-step factor $1 + L h$ imply a uniform global $O (h^{p})$ error with the stated exponential constant.

Proof

1.1givenF1F2F3F4F5

The assumption $y_{n} \in K \subset (0, \infty)^{m}$ makes every standard denominator $σ_{i} = (y_{n})_{i}$ positive, so [F3] is admissible at every step. By [F4] the local defect is bounded by $C h^{2}$ and the one-step map has factor $1 + L h$ , with constants independent of $n$ and $N$ . The initial error is zero, and the tube hypothesis supplies the common-tube condition of [F5].

2.1F5step 1.1algebra∎

Apply [F5] with $p = 1$ and $C_{0} = 0$ . If $L > 0$ , its bound is $C (\exp (L T) - 1) h / L$ ; if $L = 0$ , it is $C T h$ . Both are $C_{T} h$ for a constant independent of $N$ .

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Certified residuals bound mass drift and the exact-solve positivity margin

Statement

Let $M \in R^{m \times m}$ be the exact real matrix represented by the stored coefficients of a successfully completed Modified Patankar Euler for conservative transfer systems step with admissible positive denominators, or of a completed stage/final solve of The positive-interior MPRK22(1) step, for a finite conservative PDS. Let $b \geq 0$ be its stored right-hand side, interpreted exactly as a real vector. For either scheme, the PDS transfer identity gives $1^{T} M = 1^{T}$ and strict column diagonal dominance, so $M^{- 1} \geq 0$ . Let $\hat{y} \in R^{m}$ be a stored candidate. Assume all entries of $M$ , $\hat{y}$ , and $b$ are stored in a common operationwise format $F$ with unit roundoff $u$ , and compute $\hat{v} = fl (M \hat{y})$ and $\hat{w} = fl (\hat{v} - b)$ in the model of Operationwise real and complex floating point arithmetic with rounded square roots and fused multiply-add, with $0 < u < 1$ , $(m - 1) u < 1$ , every nonzero exact intermediate normal, zero operations exact, and successful sequential dot products and subtraction. Let $W$ be the certified allowance from Rounded matrix products give certified linear and normal-equation residual allowances, with its norm inputs exact or certified upper bounds. Let $η_{w}$ be the exact value of $∥ \hat{w} ∥_{2}$ or a certified upper bound for it, and set $ρ : = η_{w} + W$ .

$∥ M \hat{y} - b ∥_{2} \leq ρ, ∣ 1^{T} \hat{y} - 1^{T} b ∣ \leq \sqrt{m} ρ .$

If $y^{*} : = M^{- 1} b$ is the exact solution of the stored linear system, then

$∥ \hat{y} - y^{*} ∥_{1} \leq \sqrt{m} ρ .$

Consequently, the componentwise margin condition ${\hat{y}}_{i} \geq \sqrt{m} ρ$ for every $i$ certifies $y^{*} \geq 0$ . A residual bound alone does not certify that the computed candidate $\hat{y}$ is nonnegative. This certificate is relative to the stated stored matrix and right-hand side; any coefficient-assembly or right-hand-side data error relative to the intended real system must be separately bounded and added to the residual allowance.

Facts & Assumptions

Given: A finite conservative PDS matrix $M$ from the stated Patankar systems, its stored real right-hand side $b \geq 0$ , a stored candidate $\hat{y}$ , and their common operationwise format with $0 < u < 1$ and $(m - 1) u < 1$ . Assume normal nonzero exact intermediates, exact zero operations, successful residual calculations, exact or certified upper bounds for the norm inputs used in $W$ , and an exact or certified upper bound $η_{w} \geq ∥ \hat{w} ∥_{2}$ .

[F1]

Finite conservative production destruction systems: paired transfers satisfy $p_{i j} = d_{j i}$ .

[F2]

Modified Patankar Euler for conservative transfer systems and The positive-interior MPRK22(1) step: every completed solve evaluates PDS rates only at states in $Ω_{+}$ ; with positive declared denominators, its matrix has nonpositive off-diagonal entries, positive diagonal, and column sums equal to one by $p_{i j} = d_{j i}$ .

[F3]

Strict column dominance gives an entrywise nonnegative inverse: each such solve matrix has an entrywise nonnegative inverse.

[F4]

Operationwise real and complex floating point arithmetic with rounded square roots and fused multiply-add: successful real additions, subtractions and products follow the stated no-underflow, round-to-nearest operation model.

[F5]

Rounded matrix products give certified linear and normal-equation residual allowances: for stored $M, \hat{y}, b$ and the stated successful evaluation order, $∥ \hat{w} - (M \hat{y} - b) ∥_{2} \leq W$ when its unit-roundoff and normal-intermediate hypotheses hold; the norms used in $W$ are exact or certified upper bounds.

[F6]

Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥_{2} ∥ v ∥_{2}$ , in particular $∥ r ∥_{1} \leq \sqrt{m} ∥ r ∥_{2}$ and $∣ 1^{T} r ∣ \leq \sqrt{m} ∥ r ∥_{2}$ .

Proof

1.1F1F2F3algebra

For modified Patankar Euler, column $j$ of $M$ has off-diagonal entries $- h p_{i j} (y^{n}) / σ_{j}$ and diagonal $1 + h \sum_{k \neq j} d_{j k} (y^{n}) / σ_{j}$ ; for an MPRK final matrix the same form uses averaged rates and denominators $z_{j}$ , and its stage is the first form with $σ_{j} = y_{j}^{n} > 0$ . By [F1]–[F2], each such matrix has column sum one and strict column-dominance margin one, so [F3] gives $M^{- 1} \geq 0$ .

1.2F4F5algebra

Put $r : = M \hat{y} - b$ . By [F5] and the triangle inequality, $∥ r ∥_{2} \leq ∥ \hat{w} ∥_{2} + ∥ r - \hat{w} ∥_{2} \leq ∥ \hat{w} ∥_{2} + W \leq η_{w} + W = ρ$ .

1.3F2F3algebra

Multiplying $1^{T} M = 1^{T}$ on the right by $M^{- 1}$ gives $1^{T} M^{- 1} = 1^{T}$ .

2.1F2F6step 1.2algebra

Since $1^{T} M = 1^{T}$ , $1^{T} \hat{y} - 1^{T} b = 1^{T} (M \hat{y} - b) = 1^{T} r$ ; [F6] and step 1.2 give $∣ 1^{T} \hat{y} - 1^{T} b ∣ \leq \sqrt{m} ρ$ .

2.2F3F6step 1.2step 1.3algebra

With $y^{*} = M^{- 1} b$ , we have $\hat{y} - y^{*} = M^{- 1} r$ and, since $M^{- 1} \geq 0$ and its columns sum to one, $∥ M^{- 1} r ∥_{1} = \sum_{i} ∣ \sum_{j} (M^{- 1})_{i j} r_{j} ∣ \leq \sum_{j} (\sum_{i} (M^{- 1})_{i j}) ∣ r_{j} ∣ = ∥ r ∥_{1} \leq \sqrt{m} ρ$ by [F6] and step 1.2.

3.1step 2.2algebra∎

For each coordinate, $y_{i}^{*} \geq {\hat{y}}_{i} - ∣ {\hat{y}}_{i} - y_{i}^{*} ∣ \geq {\hat{y}}_{i} - ∥ \hat{y} - y^{*} ∥_{1}$ ; step 2.2 therefore gives $y_{i}^{*} \geq 0$ whenever ${\hat{y}}_{i} \geq \sqrt{m} ρ$ . A residual bound alone places no sign condition on $\hat{y}$ , so it does not certify that the computed candidate is nonnegative.

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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.

5 · Examples, counterexamples and false statements

None yet.

Positive Conservative Production Destruction Integrators

1 · Prerequisites

2 · Summary

Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

Finite conservative production destruction systems

Definition

Strict column dominance gives an entrywise nonnegative inverse

Statement

Facts & Assumptions

Proof

Modified Patankar Euler for conservative transfer systems

Definition

Source-factored transfer rates for zero compartments

Definition

Zero-compatible modified Patankar Euler in rate-factor form

Definition

The positive-interior MPRK22(1) step

Definition

Modified Patankar Euler preserves positivity and total mass

Statement

Facts & Assumptions

Proof

Modified Patankar Euler has a uniform first-order defect and one-step bound on positive tubes

Statement

Facts & Assumptions

Proof

The factorized zero-state step is well-defined, nonnegative, conservative, and locally first order

Statement

Facts & Assumptions

Proof

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

Statement

Facts & Assumptions

Proof

MPRK22(1) has a uniform second-order defect and one-step bound on positive tubes

Statement

Facts & Assumptions

Proof

Modified Patankar Euler converges globally with first-order error on a positive tube

Statement

Facts & Assumptions

Proof

Certified residuals bound mass drift and the exact-solve positivity margin

Statement

Facts & Assumptions

Proof

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

Statement

Facts & Assumptions

Proof

5 · Examples, counterexamples and false statements

Sources