docs:tutorials:findsoln
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docs:tutorials:findsoln [2014/12/04 12:40] gibson |
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| \begin{eqnarray*} | \begin{eqnarray*} | ||
| \sigma_{xyz} : [u,v,w] (x,y,z) &\rightarrow [-u,-v,-w] (-x,-y,-z) \\ | \sigma_{xyz} : [u,v,w] (x,y,z) &\rightarrow [-u,-v,-w] (-x,-y,-z) \\ | ||
| - | \sigma_{xy*} \tau_{xz} : [u,v,w] (x,y,z) &\rightarrow [-u,v,-w] (-x+L_x/2,y,-z+L_z/2) | + | \sigma_{xy} \tau_{xz} : [u,v,w] (x,y,z) &\rightarrow [-u,-v,w] (-x+L_x/2,-y,z+L_z/2) |
| - | \end{eqnarray} | + | \end{eqnarray*} |
| The pointwise inversion $\sigma_{xyz}$ of this group fixes the origin and prevents traveling waves | The pointwise inversion $\sigma_{xyz}$ of this group fixes the origin and prevents traveling waves | ||
| Line 55: | Line 55: | ||
| {{:docs:tutorials:2014-12-04-a.png?direct&600|}} | {{:docs:tutorials:2014-12-04-a.png?direct&600|}} | ||
| - | You can see a nice horizontal blue streak at $T \approx 60$ over the range $825 \leq t \leq 1000$ that indicates a long period over which $\|u(t) - u(t+60)\|$ is small. This suggests the turbulent trajectory is shadowing a periodic orbit with period $T \approx 60$ over this interval. The minimum of $\| u(t) - u(t+T) \|$ in this region occurs at $t=917$ and $T=63$. That's a good candidate for a search. | + | You can see strikingly periodic behavior over the range $825 \leq t \leq 1000$ with nice horizontal blue streak at $T \approx 65$. This suggests the turbulent trajectory is shadowing a periodic orbit with period $T \approx 65$. The minimum of $\| u(t) - u(t+T) \|$ in this region occurs at $t=917$ and $T=63$. That's an unusually promising initial guess for a periodic orbit. |
| === 4. Find the periodic orbit with ''findsoln'' === | === 4. Find the periodic orbit with ''findsoln'' === | ||
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| findsoln -orb -T 63 -dt 0.05 -R 400 -symms ../sxyz_sxytxz.asc ../data/u917.h5 | findsoln -orb -T 63 -dt 0.05 -R 400 -symms ../sxyz_sxytxz.asc ../data/u917.h5 | ||
| </code> | </code> | ||
| + | |||
| Since I tend to run many searches for many initial guesses, I like to do each search in a subdirectory | Since I tend to run many searches for many initial guesses, I like to do each search in a subdirectory | ||
| - | named after the initial guess. The ''findsoln'' command runs a Newton-Krylov-hookstep search that finds a $u,T$ solution of the equation $u - f^T(u) = 0$ given the initial guess $T=63$ and $u = u(917)$. The search is restricted to the $\langle \sigma_{xyz}, \sigma_{xz} \tau_{xz} \rangle$ symmetry group. The symmetry restriction vastly reduces the search space and results in a faster and more robust search. | + | named after the initial guess. The ''findsoln'' command runs a Newton-Krylov-hookstep search that finds a $u,T$ solution of the equation $u - f^T(u) = 0$ given the initial guess $T=63$ and $u = u(917)$. The search is restricted to the $\langle \sigma_{xyz}, \sigma_{xy} \tau_{xz} \rangle$ symmetry group. The symmetry restriction vastly reduces the search space and results in a faster and more robust search. |
docs/tutorials/findsoln.1417725653.txt.gz · Last modified: 2014/12/04 12:40 by gibson
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