Nstrophy/README.md

Nstrophy is a tool to solve the two-dimensional Navier-Stokes equation as well
as Gallavotti's reversible Navier-Stokes equation and compare them.

**Nstrophy is under active development**

# Building

Compile Nstrophy with
```bash
make
```
which will place a binary at `build/nstrophy`.

The syntax for the execution of Nstrophy is
```bash
./build/nstrophy [-p parameters] [-s savefile] [-u u_outfile] [-t nthreads] <command>
```
* `parameters` is a list of parameters for the computation, see
  [Parameters](#parameters)

* `savefile` is a file where the last step of the computation is saved in
  binary format so that the computation can be resumed after it has terminated,
  see
  [Interrupting and resuming the computation](#interrupting-and-resuming-the-computation).

* `u_outfile` is a file to which the final u is written in plain text format,
  which can be used as an initial condition for a future computation.

* `nthreads` is the number of threads used to compute Fast Fourier Transforms.

Nstrophy is written in C. The Makefile uses the GNU C Compiler.

Nstrophy depends on `fftw`: [https://fftw.org]


# Commands

The available commands are

* `enstrophy`: to compute the enstrophy and various other observables. This
  command prints
  ```step_index time average(alpha) average(energy) average(enstrophy) alpha energy enstrophy```
  where the averages are running averages over `print_freq`. In addition, if
  the algorithm has an adaptive step, an extra column is printed with `delta`.
  In addition, if alpha has a negative value (even in between `print_freq`
  intervals), a line is printed with the information.

* `lyapunov`: to compute the Lyapunov exponents. This command prints
  ```time instantaneous_lyapunov lyapunov```
  where `instantaneous_lyapunov` is computed from the tangent flow only between
  the given time and the previous one.

* `uk`: to compute the Fourier transform of the solution.

* `quiet`: does not print anything, useful for debugging.


# Parameters

The parameters can be specified using the `-p` flag. The parameter string
should be a `;` sperated list of `key=value` pairs. The possible keys are

* `equation`: either `irreverisible` (default) or `reversible`.

* `K` (int, default 16): cutoff in momentum space: -K<=k_i<=K

* `K1` (int, default `K`): cutoff in momentum space for the x component:
  -K<=k_x<=K

* `K2` (int, default `K`): cutoff in momentum space for the y component:
  -K<=k_y<=K

* `N` (int, default smallest power of 2 that is larger than 3`K`): size of fft
  vectors: must be at least 3 times `K` to avoid aliasing.

* `N1` (int, default `N`): same as `N` but only for x component.

* `N2` (int, default `N`): same as `N` but only for y component.

* `final_time` (double, default 100000): time at which to end the computation.
  Set to <0 to keep on going forever.

* `nu` (double, default 0.00048828125): viscosity.

* `delta` (double, default 0.0001220703125): step size.

* `L` (double, default 2pi): size of box.

* `print_freq` (double, default 1): only print when time crosses integer
  multiples of `print_freq`.

* `starting_time` (double, default 0): starting time.

* `driving`: either `zero` for no driving, `test` (default) for a testing
  driving force or `file:<filename>` or `file_txt:<filename>` to read the
  driving force from a file. When using `file:<filename>` the file should be
  binary, whereas with `file_txt:<filename>` it should be plaintext. The binary
  file format is `(double)(double)` for each entry of the driving force,
  excluding kx<0 and kx=0&&ky<=0. The plaintext file format is
  `kx ky real_part imag_part`.

* `init`: either `random` for a random initialization, `gaussian` (default) for
  a Gaussian initial condition or `file:<filename>` or `file_txt:<filename>` to
  read the driving force from a file. When using `file:<filename>` the file
  should be binary, whereas with `file_txt:<filename>` it should be plaintext.
  The binary file format is `(double)(double)` for each entry of the driving
  force, excluding kx<0 and kx=0&&ky<=0. The plaintext file format is
  `kx ky real_part imag_part`.

* `init_energy` (double, default is to not rescale the initial condition, is
  incompatible with `init_enstrophy`): enforce a value for the energy of the
  initial condition.

* `init_enstrophy` (double, default is to not rescale the initial condition, is
  incompatible with `init_energy`): enforce a value for the enstrophy of the
  initial condition.

* `random_seed` (int, default ): seed for random initialization.

* `algorithm`: fixed step methods: `RK4` for Runge-Kutta 4, `RK2` for
  Runge-Kutta 2.
  adaptive step methods: `RKF45` for Runge-Kutta-Fehlberg (order
  4), `RKDP54` for Runge-Kutta-Dormand-Prince (order 5), `RKBS32` for
  Runge-Kutta-Bogacki-Shampine (order 3).

* `adaptive_tolerance` (double, default 1e-11): when using an adaptive step
  method, this is the maximal allowed relative error.

* `adaptive_factor` (double, default 0.9): when using the RKF45 method, the
  step gets adjusted by `factor*delta*(tolerance/error)^(1/5)`.

* `max_delta` (double, default 1e-2): when using the adaptive step, do not
  exceet `delta_max`.

* `adaptive_cost`: cost function to use to estimate the error of the adaptive
  method: `L1` (default) for the normalized L1 norm, `k3` for the normalized L1
  norm of f_k/|k|^3, `k32` for the normalized L1 norm, `enstrophy` for the
  enstrophy, `alpha` for alpha.

* `keep_en_cst` (0 or 1, default 0): impose that the enstrophy is constant at
  each step (only really useful for the reversible equation).

* `print_alpha` (0 or 1, default 0): if this is set to 1, then whenever alpha
  is negative, its value is printed as a comment.

* `lyapunov_reset` (double, default: `print_freq`): if this is set, then, when
  computing the Lyapnuov exponents, the tangent flow will renormalize itself at
  times proportional to `lyapunov_reset`. This option is incompatible with
  `lyapunov_maxu`.

* `lyapunov_maxu` (double, default: unset): if this is set, then, when
  computing the Lyapnuov exponents, the tangent flow will renormalize itself
  whenever the norm of the tangent flow exceeds  `lyapunov_maxu`.  This option
  is incompatible with `lyapunov_reset`.

* `algorithm_lyapunov`: the algorithm used to integrate the tangent flow. Can
  be `RK4` for Runge-Kutta 4 (default) or `RK2` for Runge-Kutta 2. Adaptive
  step algorithms cannot be used for the tangent flow.


# Interrupting and resuming the computation

The `enstrophy` computation can be interrupted by sending Nstrophy the `SIGINT`
signal (e.g. by pressing `Ctrl-C`.) When Nstrophy receives the `SIGINT` signal,
it finishes its current step and writes the value of uk, either to `savefile`
if such a file was specified on the command line (using the `-s` flag), or to
`stderr`. In addition, when a `savefile` is specified it writes the command
that needs to be used to resume the computation (which essentially just sets
the appropriate `starting_time` and `init:file:<savefile>` parameters. The data
written to the `savefile` is binary.


# License
Nstrophy is released under the Apache 2.0 license.

Copyright 2017-2024 Ian Jauslin