Finite-volume upwind scheme for age renewal transport

For Linear age-structured renewal transport problem, input a finite horizon $T>0$, a cutoff $A>T$, integers $J,N\ge 1$, and use the uniform grid $a_j=j\mathrm{\Delta }a$ with $\mathrm{\Delta }a=A\mathrm{/}J$ and cells $C_j=[a_{j-1},a_j]$. Set $\mathrm{\Delta }t=T\mathrm{/}N$ and $t_k=k\mathrm{\Delta }t$ for $0\le k\le N$. Use the model rates $\mu ,\beta$ and input nonnegative initial cell masses $M_j^0$. Also input one positive-birth-quadrature setting for each cell as specified in Positive composite or capped-adaptive quadrature for renewal births; if omitted, use its default one-panel midpoint setting. Keep these settings fixed at every time step. Evaluate ${\mu }_j=\mu ((a_{j-1}+a_j)\mathrm{/}2)$. Require the transport-plus-loss CFL

$\lambda +\mathrm{\Delta }t{\mu }_{\mathrm{\infty }}\le 1,\lambda =\mathrm{\Delta }t\mathrm{/}\mathrm{\Delta }a,{\mu }_{\mathrm{\infty }}=\mathrm{\parallel }\mu {\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{.}$

If the CFL inequality fails, return status invalid-cfl without an update.

Initialize birth, quadrature-status, and outflow records as empty, and set cumulative outflow $O_h^0=0$. For each $k=0,\dots ,N-1$, put $u_j^k=M_j^k\mathrm{/}\mathrm{\Delta }a$, compute $B_h^k$ by calling Positive composite or capped-adaptive quadrature for renewal births with the current masses, model fertility $\beta$, and the selected cell settings, and set the upwind fluxes $F_0^k=B_h^k$ and $F_j^k=u_j^k$ for $1\le j\le J$. Update, for $1\le j\le J$,

$M_j^{k+1}=M_j^k-\mathrm{\Delta }t(F_j^k-F_{j-1}^k)-\mathrm{\Delta }t{\mu }_j{M}_j^k\mathrm{.}$

The right flux $F_J^k$ is absorbing outflow: record and discard the step mass $\mathrm{\Delta }t{F}_J^k$, set $O_h^{k+1}=O_h^k+\mathrm{\Delta }t{F}_J^k$, and allow no inflow from ages above $A$. Append $B_h^k$, the quadrature status, and the step outflow record, then proceed to step $k+1$. Return $(M^k{)}_{k=0}^N$ with the birth, quadrature-status, step-outflow, and cumulative-outflow records. An invalid grid or failed rate/quadrature evaluation returns failure with no partial current step. A cap-reached adaptive quadrature estimate may be used for a finite step, but its status is retained and carries no accuracy claim.