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SUMMARY:Pavel Bedrikovetsky (University of Adelaide)
DTSTART:20240918T162000Z
DTEND:20240918T180000Z
DTSTAMP:20260423T010529Z
UID:GDEq/115
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/GDEq/115/">E
 xact solutions and upscaling in conservation law systems</a>\nby Pavel Bed
 rikovetsky (University of Adelaide) as part of Geometry of differential eq
 uations seminar\n\n\nAbstract\nNumerous 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\, o
 r multi-block scale in city transport. The micro heterogeneity at lower sc
 ales 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-sc
 ale flux $f(u)$?\n\nWe 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 sev
 eral 1D problems: "smoothing" of shocks and "sharpening" of rarefaction wa
 ves into shocks due to microscale $x$- and $t$-dependencies\, flows in pie
 cewise homogeneous media. It also allows formulating an upscaling algorith
 m based on the analytical solutions and its invariant properties.\n
LOCATION:https://researchseminars.org/talk/GDEq/115/
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