Seminar talk, 18 September 2024

Speaker: Pavel Bedrikovetsky

Title: Exact solutions and upscaling in conservation law systems

Abstract:
Numerous transport processes in nature and industry are described by nxn conservation law systems u,t+f(u),x=0, u=(u1,...,un). 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.