Characteristic-cell overlap remapping for a conservative density

Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for this Lebesgue-measurable-volume branch. Work on the periodic box ${\mathbb{T}}_{\mathrm{\ell }}^d$ of Periodic tensor grids and multilinear interpolation. For measure calculations use the bounded open fundamental domain $\mathrm{\Omega }:={\prod }_{j=1}^d(0,{\mathrm{\ell }}_j)$; the period faces omitted from the half-open grid-indexing copy $Q={\prod }_{j=1}^d[0,{\mathrm{\ell }}_j)$ are Lebesgue-null by A box in ${\mathbb{R}}^n$ with parameters $a_i\le b_i$ is Lebesgue measurable of measure ${\prod }_{i<n}(b_i-a_i)$, whichever of its faces are included. For a periodic measurable set $E$, define $\mathrm{\mid }E\mathrm{\mid }:={\lambda }_d(E\cap \mathrm{\Omega })$, where ${\lambda }_d$ is the $d$-dimensional Lebesgue measure of Lebesgue measurable sets, the family $\mathcal{L}({\mathbb{R}}^n)$, and the restricted set function ${\lambda }_n$. Let $\{C_j{\}}_{j=1}^J$ be a finite measurable source-cell partition and $\{A_i{\}}_{i=1}^J$ a finite measurable arrival-cell partition of $\mathrm{\Omega }$ up to null sets, with every cell of positive volume; their grid labels use the half-open convention in $Q$. Let $\mathrm{\Phi }={\mathrm{\Phi }}_{t_{n+1},t_n}$ be the exact characteristic map from time $t_n$ to $t_{n+1}$. Its inverse images $D_i:={\mathrm{\Phi }}^{-1}(A_i)$ are the departure cells; Characteristic departure cells are measurable and have positive finite volume gives their measurability and positive finite volume, while A periodic C1 characteristic map is an orientation-preserving diffeomorphism gives the diffeomorphism and positive-Jacobian convention.

Given a source cell-average density ${\rho }_j^n$ on $C_j$, set ${\bar{\rho }}_i:=\frac{1}{\mathrm{\mid }{D}_i\mathrm{\mid }}{\sum }_{j=1}^J\mathrm{\mid }{D}_i\cap C_j\mathrm{\mid }{\rho }_j^n,{\rho }_i^{n+1}:=\frac{\mathrm{\mid }{D}_i\mathrm{\mid }}{\mathrm{\mid }{A}_i\mathrm{\mid }}{\bar{\rho }}_i\mathrm{.}$ The first formula is the exact intersection-volume remap of Intersection-volume conservative remapping from the source partition to the departure partition. The second is the characteristic volume factor that converts the average on a departure cell into the transported mass per arrival-cell volume. All intersections and volumes are exact in this mathematical definition; a geometric approximation or quadrature rule would require a separate error term.

The batch-63 PE17 definition is a draft carrier and its current body has been inspected. It defines the exact overlap weights $\mathrm{\mid }{T}_j\cap S_i\mathrm{\mid }$ and target-cell average for finite measurable partitions of a bounded finite-positive-measure domain; here take the open box $\mathrm{\Omega }$, source cells $S_j=C_j\cap \mathrm{\Omega }$, and target cells $T_i=D_i\cap \mathrm{\Omega }$. Countable Choice is the axiom used locally for Borel-to-Lebesgue measurability and the finite-volume branch. The separate batch-63 theorem that proves conservation and positivity also assumes Countable Choice, as recorded in the dependent theorem.