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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].

Depends on

Used by

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

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Sources

Stefan Kopecz and Andreas Meister, On order conditions for modified Patankar-Runge-Kutta schemes, §§2–3.1 (standard reference, not scraped)
Ernst Hairer, course notes, Chapter III: Équations Différentielles Ordinaires (standard reference, not scraped)