Tutorials

Tutorial 0 (getting started) walks the whole path once; the shorter ones after it go deeper, from a first eigenvalue problem to a GPU run, keeping a registry of runs, importing a Dedalus script, covariant Lyapunov vectors and Lagrangian tracers. Each one is a plain Python script, shown piece by piece on its page, that ends by checking its result against a published value or a closed form (an assert, with the reference in a comment). The output printed on each page is the output of that script, executed as it is shown; the table below says when it last ran and whether it passed.

Prerequisites: Installation with the tutorials extra (pip install "conatus[tutorials]": matplotlib for the figures, xarray for tutorial 16), and some familiarity with spectral methods. Run a tutorial with python docs/site/tutorials/code/<script>.py from a copy of the repository. Tutorial 2b needs a GPU; the check script of tutorial 14 (14b) needs Dedalus.

#	Tutorial	You will learn	Reference value
—	Getting started in ten minutes	ten minutes: one equation written as on paper, a simulation, a plot, one physical number, four guided changes	Cole (1951), Hopf (1950)
0	0. Getting started: convection between two plates	the whole path once: a native model, explain(), stability, the onset by critical, a simulation, the heat flux, the Methods paragraph, the run registry	Reid & Harris (1958)
1	1. A first eigenvalue problem: Orr–Sommerfeld	a native model, conditions by wall name, eigenvalues with residuals, critical parameters, verify	Orszag (1971)
2	2. A first initial-value problem: Rayleigh–Bénard 2D, CPU then GPU	a simulation on CPU, then on GPU; time steppers, CFL, the Methods paragraph	linear growth rate, CPU/GPU agreement
3	3. Symmetries and symmetry subspaces	linearizing a native model, symmetry subspaces: the same eigenvalue with half the unknowns	Reid & Harris (1958)
4	4. Nonlinear boundary-value problems and continuation: Bratu	a nonlinear boundary-value model, Newton on the exact Jacobian, pseudo-arclength continuation, folds	Bratu (1914), Gelfand (1963)
5	5. Invariant solutions: Kuramoto–Sivashinsky, L = 22	equilibria, travelling waves and periodic orbits of a native model by Newton–Krylov–hookstep, Floquet exponents	Cvitanović et al. (2010), Ding & Cvitanović (2016)
8	8. Polar geometry: a drum, a pipe, two rotating cylinders	polar and cylindrical geometry: a drum on cn.Disk, ready pipe and Taylor–Couette problems	Bessel zeros, Meseguer & Trefethen (2003), Recktenwald et al. (1993)
10	10. Bifurcations: branch switching, cycles, codimension 2, normal forms	branch switching, cycles, codimension 2, normal forms, Landau and Ginzburg–Landau	Lorenz (1963), Kuznetsov (2004), Cross & Hohenberg (1993)
11	11. Record, query, reproduce: the run registry	the run registry on a native simulation: record, certify, query, compare, reproduce bit for bit, cache only on request	heat equation, closed form
12	12. Chaos and ensembles: Lyapunov spectrum, covariant vectors, Poincaré map, basins	Lyapunov spectrum, covariant vectors, Kaplan–Yorke, Poincaré map, basins from one ensemble	Sprott (Lorenz), exact divergence
13	13. Where a global mode is born, and what the data say about it	direct and adjoint global modes, wavemaker, base-flow sensitivity; DMD in the exact norm	Chomaz (2005) closed forms, Towne et al. (2018)
14	14. Coming from Dedalus	a convection cell as a native model, then read from a Dedalus script: line-by-line report, run, native vs translated; optional check against Dedalus	Dedalus itself (same method, rounding; check script, needs Dedalus)
15	15. Covariant Lyapunov vectors of Kuramoto–Sivashinsky: tangencies and the physical modes	sim.lyapunov with covariant vectors: exponents with errors, covariance residual, tangencies, domination and the physical modes	Cvitanović et al. (2010), Ding et al. (2016)
16	16. Lagrangian tracers and a numerical dye	sim.particles and m.dye: exact velocity at the tracers, observed order in time, a dye whose mass is measured	Taylor–Green and Couette closed forms

Execution status

Script	Last execution	Time	Run
t00_first_steps.py	{.cn-pass}PASS	30.5 s	— ()
t00_getting_started.py	{.cn-pass}PASS	138.6 s	— ()
t01_orr_sommerfeld.py	{.cn-pass}PASS	29.6 s	— ()
t02a_rayleigh_benard_cpu.py	{.cn-pass}PASS	32.2 s	— ()
t02b_rayleigh_benard_gpu.py	{.cn-pass}PASS	14.5 s	— ()
t03_symmetries.py	{.cn-pass}PASS	3.5 s	— ()
t04_bratu.py	{.cn-pass}PASS	8.9 s	— ()
t05_ks_orbits.py	{.cn-pass}PASS	68.7 s	— ()
t08_polar.py	{.cn-pass}PASS	33.7 s	— ()
t10_bifurcations.py	{.cn-pass}PASS	28.7 s	— ()
t11_runs.py	{.cn-pass}PASS	26.6 s	— ()
t12_chaos_ensembles.py	{.cn-pass}PASS	165.3 s	— ()
t13_sensitivity_data.py	{.cn-pass}PASS	24.9 s	— ()
t14_from_dedalus.py	{.cn-pass}PASS	10.8 s	— ()
t14b_dedalus_check.py	{.cn-pass}PASS	21.8 s	— ()
t15_ks_covariant.py	{.cn-pass}PASS	52.4 s	— ()
t16_particles_dye.py	{.cn-pass}PASS	47.1 s	— ()

Scripts are executed by docs/site/run_tutorials.py, on a CPU and, for the GPU parts, on a GPU; only passing outputs are copied into the pages.