Finite-volume size transport with positive division quadrature

Input a partition $0=x_0<x_1<\cdots <x_J=X$ with $J\ge 1$, a time step $\mathrm{\Delta }t>0$, a nonnegative current mass vector $M^k=(M_j^k{)}_{j=1}^J$, and the coefficients and kernel from Size transport with a nonlocal division kernel. Partition $[0,X]$ into disjoint cells $C_j=[x_{j-1},x_j)$ for $j<J$ and $C_J=[x_{J-1},X]$, with widths $h_j=x_j-x_{j-1}>0$ and centers $x_j^c=(x_{j-1}+x_j)\mathrm{/}2$. Set $u_j^k=M_j^k\mathrm{/}{h}_j$, the incoming flux $F_0^k=0$, the internal and outgoing upwind fluxes $F_j^k=g(x_j)u_j^k$ for $1\le j\le J$, and $d_j=\mu (x_j^c)+\gamma (x_j^c)$.

For each parent cell $C_{\mathrm{\ell }}$, form nonnegative daughter coefficients

$A_{j\mathrm{\ell }}=\frac{1}{h_{\mathrm{\ell }}}{Q}_M(y\mapsto K_y(C_j);x_{\mathrm{\ell }-1},x_{\mathrm{\ell }},1),$

using Composite midpoint rule on a uniform grid on the parent-size interval. Since the one-panel midpoint rule is $Q_M(f;x_{\mathrm{\ell }-1},x_{\mathrm{\ell }},1)=h_{\mathrm{\ell }}f(x_{\mathrm{\ell }}^c)$, this gives $A_{j\mathrm{\ell }}=K_{x_{\mathrm{\ell }}^c}(C_j)\ge 0$. If $\mathrm{\Delta }t(g(x_j)\mathrm{/}{h}_j+d_j)>1$ for any cell, return invalid-cfl without an update. Otherwise the finite-volume update is

$M_j^{k+1}=M_j^k-\mathrm{\Delta }t(F_j^k-F_{j-1}^k)-\mathrm{\Delta }t{d}_j{M}_j^k+\mathrm{\Delta }t{\sum }_{\mathrm{\ell }=1}^J{A}_{j\mathrm{\ell }}{M}_{\mathrm{\ell }}^k\mathrm{.}$

The flux $F_J^k$ is absorbing outflow. The admissible variable-width CFL is $\mathrm{\Delta }t(g(x_j)\mathrm{/}{h}_j+d_j)\le 1$ for every cell. If a grid or any required coefficient/kernel evaluation fails, return failure without a partial update. Otherwise return $M^{k+1}$ together with the fluxes and daughter coefficients used in the step.