Density continuity can violate conservative advection balance

A universal junction rule declaring the density equal on every incident edge always conserves mass for advection with unequal edge speeds.

Use one incoming edge with speed $1$ and two outgoing edges with speeds $1$ and $2$. Mass flux on an edge is stipulated to be speed times density. Giving all incident edges density $1$ yields incoming flux $1$ and total outgoing flux $3$, so mass is not conserved. A conservative routing can instead send the full incoming flux to the speed-$1$ outgoing edge and zero to the other.

The arrows indicate the stipulated flow direction and speeds; the exact flux sums in the refutation, not the drawing, establish the failure.

\begin{center} \begin{tikzpicture}[>=stealth, every node/.style={font=\small}] \node[circle,draw] (v) at (0,0) {junction}; \node[left] (in) at (-2.1,0) {in: $v=1,\rho =1,q=1$}; \node[right] (out1) at (2.2,1.0) {out: $v_1=1,\rho =1,q_1=1$}; \node[right] (out2) at (2.2,-1.0) {out: $v_2=2,\rho =1,q_2=2$}; \draw[->] (in) -- (v); \draw[->] (v) -- (out1); \draw[->,dashed] (v) -- (out2); \end{tikzpicture} \end{center}