Lecture 7 : Numerical methods for problems with time-scale separation, part 2

What this episode covers

In this lecture, we continue the discussion of numerical methods for problems with time-scale separation. In particular, we illustrate the properties that were derived in lecture 6 with a concrete example, and we extend projective integration to kinetic equations in the diffusion limit. 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 In this movie, you will see the derivation of the Fourier transformation of a finite difference scheme for the kinetic equation. Background reading For the material in this lecture, I refer to the following papers C.W. Gear and I.G. Kevrekidis. Projective methods for stiff differential equations: Problems with gaps in their eigenvalue spectrum. SIAM Journal on Scientific Computing 24(4):1091-1106, 2003. 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. E. Vanden-Eijnden. Numerical techniques for multi-scale dynamical systems with stochastic effects. Comm. Math. Sci. 1:385-391 (2003). W. E, Analysis of the heterogeneous multiscale method for ordinary differential equations. Comm. Math. Sci. 1:423–436, 2003. P. Lafitte and G. Samaey, Asymptotic-preserving projective integration schemes for kinetic equations in the diffusion limit, 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.

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