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====== Continuation ====== 2009-2-13 Continuation is finding a solution to an equation by using a solution to a nearby equation as an initial guess. E.g **poor man's continuation**: Let ''u'' be a solution to the equation ''f(u,p) = 0'' with parameters ''p''. To find a solution of ''f(u',p') = 0'' with parameters ''p' = p + dp,'' take ''u'' as an initial guess and solve by Newton search. Even better is **quadratic continuation**: Take few solutions ''u'' for a few values of ''p'', make a local quadratic approximation to ''u(p)'', and then extrapolate to the parameter value you want to get a good initial guess ''u(p')''. Of course you can do linear, cubic, etc., too. Jonathon and I did our continuation mostly the poor man's way, with ''p'' usually being the Reynolds number. Sometimes I did linear continuation using the ''addfields'' utility. Jonathan wrote a ''branchswitch.cpp'' utility that tracked a solution around a saddle-node bifurcation using dissipation as the continuation parameter and producing the estimated Reynolds number for the initial guess. We also continued EQBs and POs in cell size ''(Lx, Lz)'' or equivalently (α,γ) = (2π/Lx, 2π/Lz). This is more troublesome because you have to satisfy the divergence constraint. Typically I would use ''changegrid'' to change the cell size and fix the divergence of the stretched field. The combination of procedures we used was somewhat cumbersome and didn't help us continue around saddle-node bifurcations in spatial parameters. Because of these limitations and because I anticipate people in the study group wanting to continue solutions, I wrote a better algorithm that does quadratic continuation in an arbitrary parameter and handles geometric continuation seamlessly. This allowed me to continue EQ1 and EQ2 (Nagata LB and UB, blue in figure) around the saddle-nodes at γ=1.7ish and EQ5 (green) around the saddle-node at γ=2.9ish. {{n00bsie_gamma1.png?200}} {{n00bsie_gamma2.png?230}} **Left** as of last fall, **Right** same fig with new continuations. The figure shows energy dissipation D of plane Couette equilibria as a function of γ=2π/Lz, with α=2π/Lx=1.14 fixed and Re=400. EQ1,2 blue; EQ3,4 red; EQ5,6 green, EQ7,8 black; EQ10,11 magenta.
