gtspring2009:gibson:continuation

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gtspring2009:gibson:continuation [2009/04/06 09:25] gibson |
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as an initial guess. | as an initial guess. | ||

- | E.g **poor man's continuation**: Let ''u'' be a solution to the equation ''f(u,p) = 0'' with | + | 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,'' | 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. | take ''u'' as an initial guess and solve by Newton search. | ||

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Managed to complete the continuation of P19p02 in Re via the ''continuefields'', ''continueparams'', and ''findorbit'' utilities as shown above. It took a lot of work (running three or four utilities on PACE for each data point), way too much to do for the whole set of periodic orbits in HKW. Last night I wrote a Matlab arclength continuation code that can branch-switch through pitchfork and saddle-node bifurcations. It'll take a little time to reorganize ''findorbit'' as a function to be called from a continuation routine, but it'll be worth it. | Managed to complete the continuation of P19p02 in Re via the ''continuefields'', ''continueparams'', and ''findorbit'' utilities as shown above. It took a lot of work (running three or four utilities on PACE for each data point), way too much to do for the whole set of periodic orbits in HKW. Last night I wrote a Matlab arclength continuation code that can branch-switch through pitchfork and saddle-node bifurcations. It'll take a little time to reorganize ''findorbit'' as a function to be called from a continuation routine, but it'll be worth it. | ||

+ | **2009-09-16 Predrag:** This is great! | ||

===== 2009-04-06 HKW orbit continuation ===== | ===== 2009-04-06 HKW orbit continuation ===== | ||

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nodes. (2 x 20ish solutions is 40ish computations). Continuation still requires manual intervention now and | nodes. (2 x 20ish solutions is 40ish computations). Continuation still requires manual intervention now and | ||

then. Anyone is interested in using this utility please let me know and I will describe its ins and outs. | then. Anyone is interested in using this utility please let me know and I will describe its ins and outs. | ||

+ | ===== 2009-09-16 ===== | ||

+ | |||

+ | Recently resumed continuation of these orbits for a PO paper, to be produced a.s.a.p., to demonstrate parametric robustness of these solutions. Significantly improved continuation algorithm by restricting continuation point to I-D=0 Poincare section to eliminate temporal phase shift. Contrary to my guess there are still occasional problems with tangency to this section. Below is a portion of the D vs I graph for the P46p23 orbit at three successive Reynolds numbers. You can see that the intersection of the orbit with I=D line goes tangent somewhere between the red and green orbits. My algorithm tries to extrapolate in Re the point of intersection of the orbit with the I=D line, so it gets stuck when the orbit goes tangent to the line. Bummer! Fortunately this doesn't happen very often and there is an easy diagnostic for detecting when you get close to tangency, at which point you can switch to a more transverse intersection. I will need to add this to my continuation code, if I am lucky it will explain a couple stuck continuations I have at the moment and get them going again. | ||

+ | |||

+ | {{:gtspring2009:gibson:hkw:2009-09-16-a.png?300}} {{:gtspring2009:gibson:hkw:2009-09-16-b.png?300}} | ||

+ | |||

+ | **2009-09-16 Predrag:** There are 2 or 4 D-I=0 sections. Can't you simply switch search to another section whenever a pair of periodic points (in the section) is getting too close? Typical distance on diagonal to the next periodic point that cycles/pivots around the mean <D> is of order 1, so whenever the next periodic point is on the same side of <D> , distance less than 0.1, simply switch the continuation to the next cycle point? | ||

+ | |||

+ | **2009-09-17 JFG:** Yes, agreed, switching to a different intersection is the thing to do. My practical question is to what extent right now should I try to automate the switchover in my continuation code, versus simply warning when the condition gets bad and stopping the continuation, and then personally examining D vs I plots, choosing a new intersection, and restarting the continuation. In the long run complete automation will be probably be worthwhile, but I always like to do something manually a few times before trying to automate it, to get a robust sense of the problem and a better idea of the cost of implementing automation robustly versus occasional manual intervention. | ||

+ | |||

+ | **2009-09-17 Predrag:** Go manual - you want to continue only a few shortest periodic orbits, not all longer ones still to come. | ||

+ | |||

+ | **2009-09-19 JFG:** Agreed. | ||

+ | ===== 2009-09-23 ===== | ||

+ | |||

+ | Some new results for continuation of orbits in Re, aspect ratio, and Lx,Lz diagonal. These were generated using continuation on the I-D=0 Poincare section, and temporal integration constrained to the s1,s2 invariant subspace. | ||

+ | (So few words, so much work...) | ||

+ | |||

+ | {{:gtspring2009:gibson:continuation:2009-09-23-a.png?300}}{{:gtspring2009:gibson:continuation:2009-09-23-b.png?300}} | ||

+ | |||

+ | {{:gtspring2009:gibson:continuation:2009-09-23-c.png?300}}{{:gtspring2009:gibson:continuation:2009-09-23-d.png?300}} | ||

+ | |||

+ | The aspect-ratio and diagonal continuations are still running, as are a couple in Re. Aspect-ratio continuations hold the diagonal sqrt(Lx^2 + Lz^2) fixed, and vice versa. Unfortunately I messed up the way the diagonal was held fixed, so I have to recompute the aspect-ratio plots. Expect that plot to change. I think these results are wonderful; they show us that the short orbits are robust in parameters, and that most of them so far are born in saddle-node bifurcations around Re=320. The orbits are surprisingly robust in aspect ratio and diagonal length. The above plots use mean dissipation on the vertical axis, but even more interesting will be the sum of the unstable eigenvalues or | ||

+ | product of the unstable multipliers. | ||

+ | |||

+ | **2009-09-25** I determined that Ny=49 is overkill for these computations using ''resolution.cpp'' (which reports the magnitude of the largest truncated coefficients and recomputes the solution residual on a finer grid --it is amazing what a difference it make to my daily work to have tools such as this packaged and made easy to use), so I restarted the continuation calculations with Ny=33. But some have not restarted cleanly, indicating other resolution problems, namely in Nx and Nz. Of course changing the box size, as you do in aspect ratio and diagonal continuation will call for changing the resolution. It just happened sooner than I expected. Am investigating, will report mroe later. | ||

+ | |||

+ | |||

+ | ===== 2009-10-01 ===== | ||

+ | |||

+ | **2009-10-01** More continuations. Have determined that P41p36, P30p99, P31p17, and P31p81 are all connected. P19p02 terminates in a Hopf bifurcation at around Lx/Lz=1.6 | ||

+ | |||

+ | {{:gtspring2009:gibson:continuation:2009-10-02-a.png?300}} | ||

+ | {{:gtspring2009:gibson:continuation:2009-10-02-b.png?300}} {{:gtspring2009:gibson:continuation:2009-10-02-c.png?300}} | ||

+ | |||

+ | {{:gtspring2009:gibson:continuation:2009-10-02-d.png?300}} | ||

+ | {{:gtspring2009:gibson:continuation:2009-10-02-e.png?300}} {{:gtspring2009:gibson:continuation:2009-10-02-f.png?300}} | ||

+ | |||

+ | ===== EQBs in 2pi x 1pi box ===== | ||

+ | |||

+ | {{:gtspring2009:gibson:continuation:2010-01-19a.png?400}} {{:gtspring2009:gibson:2010-01-18b.png?400}} | ||

+ | |||

+ | {{:gtspring2009:gibson:continuation:eq1_2pi1pi.png?300}} {{:gtspring2009:gibson:continuation:eq2_2pi1pi.png?300}} {{:gtspring2009:gibson:continuation:eq4_2pi1pi.png?300}} | ||

+ | |||

+ | {{:gtspring2009:gibson:continuation:eq7_2pi1pi.png?300}} {{:gtspring2009:gibson:continuation:eq8_2pi1pi.png?300}} {{:gtspring2009:gibson:continuation:eq9_2pi1pi.png?300}} | ||

+ | |||

+ | |||

+ | In preparation for continuing with pressure gradient for near-wall solutions, I continued some of our alpha,gamma = 1.14,2.5 EQB solutions to alpha,gamma = 1,2 or Lx,Lz = 2pi,1pi. There was really no particular sense to that box. Lx,Lz = 2pi,1pi is no better physically, but it is more conventional. EQ3 does not exist for this box; EQ4 connects to EQ7 near //Re=200// instead. I did not try to continue other EQBs. I am trying to divide my work between near-wall solutions, localized solutions, and the //I-D=0// Poincare map for the HKW orbits. Could use some grad students. | ||

+ |

gtspring2009/gibson/continuation.txt ยท Last modified: 2010/02/02 07:55 (external edit)