# 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 conservation law systems , . 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 - or -dependencies into the large-scale conservation law system, like or . 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 -dependent to calculate the upper-scale flux ?

We present general case for and several systems for and . 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 - and -dependencies, flows in piecewise homogeneous media. It also allows formulating an upscaling algorithm based on the analytical solutions and its invariant properties.