Coagulation Fragmentation and Sectional Population Balance Numerics — Examples

The daughter mass sum is $2a+3a=5a$, so this daughter law satisfies the required parent-mass normalization in [F1].

Its number increment is $1+1-1=1$. Its mass increment is $-5a+2a+3a=0$, agreeing with [F2].

With $\phi (x)=x$, the coagulation increment is $(x+y)-x-y=0$. The integral equation therefore gives $M_1({\mu }_t)=M$ on the solution interval.

With $\phi =1$, the increment is $1-1-1=-1$, so $N^{\mathrm{\prime }}(t)=-\kappa N(t{)}^2\mathrm{/}2$. If $N_0>0$, then on every interval where $N>0$, differentiating $1\mathrm{/}N(t)$ gives $(1\mathrm{/}N{)}^{\mathrm{\prime }}=\kappa \mathrm{/}2$, hence $N(t)=N_0\mathrm{/}(1+\kappa N_{0}t\mathrm{/}2)$. This expression is positive at every finite $t\ge 0$, so continuity precludes a first finite zero and the formula holds throughout the solution interval. If $N_0=0$, positivity gives $N(t)\ge 0$ and the differential equation gives $N^{\mathrm{\prime }}(t)\le 0$, so $N(t)=0$, again the formula.

Substituting $x_i=2$, $x_{i+1}=4$, and $z=3$ in [F1] gives both weights equal to $(4-3)\mathrm{/}(4-2)=1\mathrm{/}2$. Thus the replacement is exactly $\frac{1}{2}{\delta }_2+\frac{1}{2}{\delta }_4$.

The weights sum to $1$, their weighted mass is $(2+4)\mathrm{/}2=3$, and their weighted second moment is $(2^2+4^2)\mathrm{/}2=10$. Hence the number and mass statements agree with [F2], and the second-moment excess is $10-3^2=1$.

On a grid with top pivot $N$, the state with $n_N=1\mathrm{/}N$ and all other retained coordinates zero has first moment $N(1\mathrm{/}N)=1$ and second moment $N^2(1\mathrm{/}N)=N$. Its reservoir may be zero, so the conserved augmented mass is $1$. As $N$ increases these admissible mass-one states have unbounded second moment, proving the finite mass identity alone gives no cutoff-independent moment bound.

The continuum product-kernel solution in [F2] has $M_2(t)=1\mathrm{/}(1-t)$, which diverges at the finite time $1$ while its first moment remains $1$. The kernel $K(x,y)=xy$ is superlinear and unbounded, outside the bounded-kernel global theorem's hypotheses; therefore this example warns against inferring continuum mass-conserving global moment control from finite-grid conservation.

For masses $2$ and $3$, their sum is $5$, which exceeds $R=4$. The rule [F1] therefore consumes masses $2+3$ and increments the reservoir by $5$.

The retained mass change is $-(2+3)=-5$ and the reservoir change is $+5$, so augmented mass change is zero, as asserted in [F2]. This reservoir entry has no later fragmentation dynamics under [F1].