Exact solutions and upscaling in conservation law systems

Numerous transport processes in nature and industry are described by $n\times n$ conservation law systems $u_t+f(u)_x=0$, $u=(u^1,\dots,u^n)$. This corresponds to upper scale, like rock or core scale in porous media, column length in chemical engineering, or multi-block scale in city transport. The micro heterogeneity at lower scales introduces $x$- or $t$-dependencies into the large-scale conservation law system, like $f=f(u,x)$ or $f(u,t)$. Often, numerical micro-scale modelling highly exceeds the available computational facilities in terms of calculation time or memory. The problem is a proper upscaling: how to "average" the micro-scale $x$-dependent $f(u,x)$ to calculate the upper-scale flux $f(u)$?
We present general case for $n=1$ and several systems for $n=2$ and $3$. The key is that the Riemann invariant at the microscale is the "flux" rather than "density". It allows for exact solutions of several 1D problems: "smoothing" of shocks and "sharpening" of rarefaction waves into shocks due to microscale $x$- and $t$-dependencies, flows in piecewise homogeneous media. It also allows formulating an upscaling algorithm based on the analytical solutions and its invariant properties.