Positivity, L1 growth and data stability of the renewal finite-volume step

In midpoint mode, each of the $q_j$ samples has weight $\mathrm{\Delta }a\mathrm{/}{q}_j>0$, so the weights sum to the cell width. In capped-adaptive mode, every returned estimate is a finite sum of Simpson panel estimates on its leaf intervals; a two-panel Simpson block has weights proportional to $(1,4,1)$, all positive, and their sum is the block width. The leaf intervals partition the cell, so the combined weights are nonnegative and sum to its width, including when a cap is reached. Hence the cell contribution lies between $0$ and $u_j{\beta }_{\mathrm{\infty }}\mathrm{\Delta }a={\beta }_{\mathrm{\infty }}{M}_j$, and summing over cells gives the stated birth bound. A failure returns no estimate.

Substituting $F_j^k=M_j^k\mathrm{/}\mathrm{\Delta }a$ in [F2] gives $M_1^{k+1}=(1-\lambda -\mathrm{\Delta }t{\mu }_1)M_1^k+\mathrm{\Delta }t{B}_h^k$ and, for $j\ge 2$, $M_j^{k+1}=(1-\lambda -\mathrm{\Delta }t{\mu }_j)M_j^k+\lambda M_{j-1}^k$. The CFL makes every displayed coefficient nonnegative, while [F1] makes $B_h^k\ge 0$; induction in $k$ proves nonnegativity.

Summing [F2] over cells telescopes the fluxes to $F_J^k-F_0^k$, hence $N_h^{k+1}=N_h^k-\mathrm{\Delta }t{F}_J^k+\mathrm{\Delta }t{B}_h^k-\mathrm{\Delta }t{\sum }_j{\mu }_j{M}_j^k$. Since $F_J^k,{\mu }_j,M_j^k\ge 0$ and step 1.1 gives $B_h^k\le {\beta }_{\mathrm{\infty }}{N}_h^k$, the asserted mass-growth inequality follows.

The update is a linear matrix with nonnegative entries by step 1.2. In each column, transport and mortality leave total coefficient at most $1$, because outflow or mortality removes mass; the birth quadrature adds at most $\mathrm{\Delta }t{\beta }_{\mathrm{\infty }}$ to that column by step 1.1. Thus its induced ${\mathrm{\ell }}^1$ norm is at most $1+\mathrm{\Delta }t{\beta }_{\mathrm{\infty }}$. Applying this matrix to $M^k-{\tilde{M}}^k$ proves the difference estimate.