docs:utils:findsoln
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docs:utils:findsoln [2010/03/04 08:41] gibson |
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| ====== findsoln ====== | ====== findsoln ====== | ||
| - | find an equilibrium, traveling wave, periodic orbit, or relative periodic orbit of plane Couette or channel flow | + | find an equilibrium, traveling wave, periodic orbit, or relative periodic orbit of plane Couette or channel flow, |
| + | using a [[docs:math:newton_krylov_hookstep| Newton-Krylov-hookstep search algorithm]]. | ||
| ====== main options ====== | ====== main options ====== | ||
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| ====== solution type ====== | ====== solution type ====== | ||
| - | ''findsoln'' finds solutions of //σ f^T(u) - u = 0// where //f^T// is the time-//T// forward map of the Navier-Stokes equations plus boundary conditions and σ is a symmetry of the flow. This equation has four kinds of solutions | + | ''findsoln'' finds solutions of //σ f^T(u) - u = 0// where //f^T// is the time-//T// forward map of the Navier-Stokes equations plus boundary conditions and σ is a symmetry of the flow. This equation has several kinds of solutions |
| * equilibrium: //σ = 1///, all //T// | * equilibrium: //σ = 1///, all //T// | ||
| - | * relative equilibrium: //σ = // a translation that scales linearly in //T//, all //T// (also called a traveling wave) | + | * relative equilibrium: //σ = // a translation or phase shift that scales linearly in //T//, all //T// (also called a traveling wave) |
| * periodic orbit: //σ = 1// and some finite value of //T// (and integer multiples of it) | * periodic orbit: //σ = 1// and some finite value of //T// (and integer multiples of it) | ||
| * preperiodic orbit: //σ ≠ 1// but //σ^n = 1// for some //n>1//, and some finite value of //T// (and integer multiples) | * preperiodic orbit: //σ ≠ 1// but //σ^n = 1// for some //n>1//, and some finite value of //T// (and integer multiples) | ||
| * relative periodic orbit: some element //σ// of the appropriate symmetry group, and some finite value of //T// (and integer multiples) | * relative periodic orbit: some element //σ// of the appropriate symmetry group, and some finite value of //T// (and integer multiples) | ||
| - | The preperiodic orbit is a portion of a periodic orbit, so I haven't really counted it as a separate solution type. | + | You specify which kind of solution to find by telling the search algorithm which variables to treat as unknowns, with the ''-eqb, -orb, -xrel, -zrel'' options. The ''-eqb'' option specifies that //T// is held fixed; the ''-orb'' option specifies that is unknown. The ''-xrel'' and ''-zrel'' options specifying that the //x// and //z// phase shifts in σ are unknown. The appropriate combinations for each solution type are best shown by example. |
| - | You specify which kind of solution to find with the <code> -eqb, -orb, -xrel, -zrel</code> options. The symmetry σ for the relative equilibrium and relative periodic orbit solutions involves unknown phase shifts (translations) in //x//, //z//, or both //x,z//, depending on the symmetry of the initial guess //u//, and you specify which to treat as unknowns by using the ''-xrel'' and ''-zrel'' options. Here are some examples of searches of various types | + | |
| ^command^search type^ | ^command^search type^ | ||
docs/utils/findsoln.1267720867.txt.gz · Last modified: 2010/03/04 08:41 by gibson
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