Lecture 6 : Numerical methods for problems with time-scale separation

What this episode covers

In this lecture, we discuss numerical methods for problems with time-scale separation. In particular, we introduce Projective integration for problems with fast slaving; Coarse projective integration for problems with fast slaving; Heterogeneous multiscale methods for problems that require averaging. 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 these movies, you will see the derivations that were done on the blackboard in this lecture : derivation of the approximation error in operator splitting; derivation of the number of inner steps required in projective integration; Additionally, I gave some extra explanation on the choice of method parameters for projective integration on the blackboard. As I feel a concrete example would make this explanation much more accessible, I decided not to make a handwritten video for that, but to come back to this at the beginning of lecture 7. 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. 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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