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First-order finite-volume convergence for compatible smooth renewal data

Statement

Fix $T > 0$ and a cutoff $A > T$ . Let $n^{A}$ be the finite-age solution from Characteristic solution and mass bound for the finite-age renewal model with initial density $n_{0}^{A} \in C^{2} ([0, A])$ and assume $μ, β \in C^{2} ([0, A])$ are nonnegative and bounded. Assume $n^{A}$ extends to $C^{2} ([0, T] \times [0, A])$ . Its smoothness at the renewal corner includes $n_{0}^{A} (0) = B_{A} (0)$ and $- (n_{0}^{A})^{'} (0) - μ (0) n_{0}^{A} (0) = B_{A}^{'} (0)$ ; no boundary value is imposed at the absorbing outflow. Take integers $J, N \geq 1$ , set $Δ a = A / J$ and $Δ t = T / N$ , and use the corresponding uniform age and time grids. Initialize with exact cell masses and use the one-panel midpoint rule in each cell from Positive composite or capped-adaptive quadrature for renewal births. Assume $Δ t / Δ a \in [c_{-}, c_{+}]$ for fixed $0 < c_{-} \leq c_{+} < \infty$ and the transport-plus-loss CFL. Define

$ϵ_{Q} : = \max_{0 \leq k < N} ∣ B_{h} (\overline{M} (t_{k})) - \int_{0}^{A} β (a) n^{A} (t_{k}, a) d a ∣,$

where $B_{h} (\overline{M} (t_{k}))$ is the birth-rate quadrature applied to the exact cell-average masses ${\overline{M}}_{j} (t_{k}) = \int_{C_{j}} n^{A} (t_{k}, a) d a$ . Then the cell-mass error $E_{k} = \sum_{j} ∣ M_{j}^{k} - \int_{C_{j}} n^{A} (t_{k}, a) d a ∣$ obeys

$\max_{0 \leq k \leq N} E_{k} \leq C_{T} (Δ a + Δ t + ϵ_{Q}) .$

The piecewise-constant reconstruction $u_{h}^{k} = M_{j}^{k} / Δ a$ on cell $C_{j}$ and zero above $A$ satisfies the same $L^{1}$ error order. For the midpoint rule, $ϵ_{Q} \leq C_{T} Δ a$ under the stated regularity, so the method is first order in age mesh and time step. If the capped adaptive rule is used instead, the estimate requires an independently established true $ϵ_{Q}$ ; its reported indicator is not substituted for that hypothesis.

Facts & Assumptions

Given: The smooth finite-cutoff solution and finite-volume scheme from Finite-age renewal model with absorbing outflow, Characteristic solution and mass bound for the finite-age renewal model and Finite-volume upwind scheme for age renewal transport, under the hypotheses in the Statement.

[F1]

Finite-age renewal model with absorbing outflow: $n_{t} + n_{a} = - μ n$ on $0 < a < A$ , the renewal inflow is $B_{A} (t) = \int_{0}^{A} β (a) n^{A} (t, a) d a$ , and the upper boundary is absorbing outflow.

[F2]

Characteristic solution and mass bound for the finite-age renewal model: $n^{A}$ follows characteristics, satisfies the renewal relation, and its exact initial-cohort outflow is absorbing.

[F3]

Finite-volume upwind scheme for age renewal transport: the cell-mass step is $M_{j}^{k + 1} = M_{j}^{k} - Δ t (F_{j}^{k} - F_{j - 1}^{k}) - Δ t μ_{j} M_{j}^{k}$ with upwind face fluxes and absorbing top outflow.

[F4]

Positivity, L1 growth and data stability of the renewal finite-volume step: for the same coefficients and positive quadrature, a one-step mass difference is amplified by at most $1 + Δ t β_{\infty}$ .

[F5]

Composite midpoint error bound for a twice continuously differentiable function: for $f \in C^{2} [a, b]$ , the composite midpoint error on width $h$ is at most $(b - a) h^{2} \max ∣ f^{''} ∣ / 24$ .

[F6]

Finite geometric error bound from a one-step recurrence: if $e_{0} \leq C_{0} h$ and $e_{k + 1} \leq (1 + L h) e_{k} + C h^{2}$ on a fixed grid $N h = T$ , then $\max_{k} e_{k} \leq C_{T} h$ .

Proof

1.1F1F2constructalgebra

Let ${\overline{M}}_{j} (t) = \int_{C_{j}} n^{A} (t, a) d a$ and ${\overline{u}}_{j} (t) = {\overline{M}}_{j} (t) / Δ a$ . The smooth finite-age characteristic solution [F2] satisfies the transport equation and renewal boundary in [F1]. Integrating [F1] over one cell and one time step gives ${\overline{M}}_{j} (t_{k + 1}) = {\overline{M}}_{j} (t_{k}) - \int_{t_{k}}^{t_{k + 1}} [n^{A} (s, a_{j}) - n^{A} (s, a_{j - 1})] d s - \int_{t_{k}}^{t_{k + 1}} \int_{C_{j}} μ (a) n^{A} (s, a) d a d s$ , with $n^{A} (s, a_{0}) = B_{A} (s)$ and the outgoing trace at $a_{J} = A$ .

2.1givenF5step 1.1algebra

Taylor's formula in age gives ${\overline{u}}_{j} (t) - n^{A} (t, a_{j}) = - \frac{Δ a}{2} (n^{A})_{a} (t, a_{j}) + O (Δ a^{2})$ uniformly. For interior cells, subtracting adjacent face-flux errors cancels the leading term and leaves $O ((Δ a)^{2})$ per cell because $(n^{A})_{a a}$ is bounded. In the first and last cells there is only one uncancelled endpoint face error, of size $O (Δ a)$ ; these two cells contribute $O (Δ t Δ a)$ in total. Taylor's formula in time gives a face-integral remainder $O (Δ t^{2} Δ a)$ per cell after adjacent differences, using bounded $(n^{A})_{t a}$ , hence $O (Δ t^{2})$ after summing. At the inlet, the smooth trace gives $(n^{A})_{t} (t, a_{0}) = B_{A}^{'} (t)$ , so the time derivative of $n^{A} (t, a_{1}) - B_{A} (t)$ is also bounded by $Δ a ∥ (n^{A})_{t a} ∥_{\infty}$ ; thus the boundary birth flux has no additional O(\Delta t^2) term beyond this face remainder and $Δ t ϵ_{Q}$ . For the reaction term, first note that ${\overline{u}}_{j} (t) = n^{A} (t, a_{j}^{c}) + O ((Δ a)^{2})$ by the symmetric cell average. Thus replacing $μ (a) n^{A} (t, a)$ by $μ (a_{j}^{c}) {\overline{u}}_{j} (t)$ has per-cell error $O ((Δ a)^{3})$ : the midpoint formula [F5] bounds the error for $μ n^{A}$ , and the cell-average replacement changes the midpoint value by $O ((Δ a)^{2})$ times the cell width. Summing and multiplying by $Δ t$ gives $O (Δ t (Δ a)^{2})$ . Freezing the reaction term at $t_{k}$ instead of integrating in time contributes $O (Δ t^{2})$ after summing cells, since $(μ n^{A})_{t}$ is bounded. Therefore $∥ ρ^{k} ∥_{ℓ^{1}} \leq C Δ t (Δ a + Δ t + ϵ_{Q})$ for sufficiently fine meshes with $Δ a \leq 1$ .

3.1F3F4step 2.1algebra

Set $e_{k} = \sum_{j} ∣ M_{j}^{k} - {\overline{M}}_{j} (t_{k}) ∣$ . Apply the one-step stability [F4] to the numerical and exact cell-average states and add the local defect from step 2.1; this yields $e_{k + 1} \leq (1 + β_{\infty} Δ t) e_{k} + C Δ t (Δ a + Δ t + ϵ_{Q})$ , with $e_{0} = 0$ .

4.1F5F6step 3.1algebra∎

Put $D = Δ a + Δ t + ϵ_{Q}$ . Iterating the recurrence in step 3.1 gives $e_{k} \leq C Δ t D \sum_{r = 0}^{k - 1} (1 + β_{\infty} Δ t)^{r} \leq C_{T} D$ , where the bound also holds when $β_{\infty} = 0$ , in which case the sum is $k \leq T / Δ t$ . For the midpoint birth rule, split each cell defect into ${\overline{u}}_{j} [Q_{M} (β; C_{j}, 1) - \int_{C_{j}} β]$ and $\int_{C_{j}} β (a) ({\overline{u}}_{j} - n^{A} (t_{k}, a)) d a$ . The first term is $O ((Δ a)^{3})$ per cell by [F5] and bounded $n^{A}$ ; in the second, subtract $β$ at the cell midpoint because $\int_{C_{j}} ({\overline{u}}_{j} - n^{A}) = 0$ , then use $∣ β (a) - β (a_{j}^{c}) ∣ \leq Lip (β) Δ a$ and $\int_{C_{j}} ∣ {\overline{u}}_{j} - n^{A} ∣ = O ((Δ a)^{2})$ from bounded $(n^{A})_{a}$ . Summing gives $ϵ_{Q} = O ((Δ a)^{2})$ , hence in particular $O (Δ a)$ . With the two-sided mesh ratio, $D = O (Δ t)$ ; the recurrence then has additive term $C^{'} Δ t^{2}$ , and [F6] gives the stated first-order estimate. Finally, on each cell $\int_{C_{j}} ∣ n^{A} (t_{k}, a) - {\overline{u}}_{j} (t_{k}) ∣ d a \leq (Δ a)^{2} ∥ (n^{A})_{a} ∥_{\infty}$ ; summing over $A / Δ a$ cells adds only $O (Δ a)$ to the cell-mass error, proving the reconstruction claim.

Depends on

Used by

Age-cutoff survival, outflow and infinite-age tail bounds Theorem

Dependency tree · two levels

27 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Benoît Perthame, Introduction to Structured Equations in Biology, full lecture notes (standard reference, not scraped)
MATH 575 lecture notes, §9 Finite difference methods for the transport equation (standard reference, not scraped)
MATH 373 lecture notes, §8 Approximation of Integrals (standard reference, not scraped)
Lecture 14: Quadrature, Introduction to Numerical Computation II, Kent State University notes (standard reference, not scraped)