Lecture 8 : Numerical methods for problems with model-scale separation

What this episode covers

In this lecture, we discuss numerical methods for problems with model-scale separation. In particular, we develop the lifting and restriction operators that are used in coarse time-stepper. We discuss lifting from density to distributions for kinetic descriptions; from moments to an ensemble of realizations from a probability distribution for stochastic differential equations (SDEs); from moments to realizations in jump processes in chemical kinetics. We also discuss the effect of noise generated at the macroscopic level during restriction. If the movie does not display nicely in your browser, here is a direct link to the movie file. Slides The slides can be found here. Additional video material Part of the lecture was done on the blackboard, namely derivation of the lattice Boltzmann method starting from a kinetic equation; derivation of a jump process for chemical reactions on a surface and its mean field limit. Background reading For the material in this lecture, I refer to the following papers I.G. Kevrekidis and G. Samaey, Equation-free modeling, Scholarpedia, 5(9):4847, 2010. I.G. Kevrekidis and G. Samaey, Equation-free multiscale computation: algorithms and applications, Annual Review of Physical Chemistry 60:321-344, 2009. P. Van Leemput, W. Vanroose and D. Roose, Mesoscale analysis of the equation-free constrained runs initialization scheme, Multiscale Modeling and Simulation 6(4):1234-1255, 2007. G. Samaey, T. Lelièvre and V. Legat, A numerical closure approach for kinetic models of polymeric fluids: exploring closure relations for FENE dumbbells, Computers and Fluids, 2010. In press. A. Provata, J.W. Turner and G. Nicolis, Nonlinear chemical dynamics in low dimensions: An exactly soluble model, Journal of Statistical Physics 70(5):1195-1213, 1993. K. Debrabant and G. Samaey, A micro/macro algorithm to accelerate Monte Carlo simulation of stochastic differential equations, 2010. Submitted. Disclaimer : These notes are provided for the benefit of participants in this course. If you are not participating in the course, and finding these pages useful, please drop me an email so I can assess the impact of this site.

Distinct summary based on available episode metadata or transcript content.