Seminar talk, 18 September 2024: Difference between revisions
Created page with "{{Talk | 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 int..." |
No edit summary |
||
| Line 2: | Line 2: | ||
| speaker = Pavel Bedrikovetsky | | speaker = Pavel Bedrikovetsky | ||
| title = Exact solutions and upscaling in conservation law systems | | title = Exact solutions and upscaling in conservation law systems | ||
| abstract = Numerous transport processes in nature and industry are described by | | abstract = Numerous transport processes in nature and industry are described by <math>n\times n</math> conservation law systems <math>u_t+f(u)_x=0</math>, <math>u=(u^1,\dots,u^n)</math>. 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 <math>x</math>- or <math>t</math>-dependencies into the large-scale conservation law system, like <math>f=f(u,x)</math> or <math>f(u,t)</math>. 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 <math>x</math>-dependent <math>f(u,x)</math> to calculate the upper-scale flux <math>f(u)</math>? | ||
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. | We present general case for <math>n=1</math> and several systems for <math>n=2</math> and <math>3</math>. 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 <math>x</math>- and <math>t</math>-dependencies, flows in piecewise homogeneous media. It also allows formulating an upscaling algorithm based on the analytical solutions and its invariant properties. | ||
| video = | | video = | ||
| slides = | | slides = | ||
Revision as of 16:21, 19 August 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.
