191 lines
7.4 KiB
Markdown
191 lines
7.4 KiB
Markdown
Nstrophy is a tool to solve the two-dimensional Navier-Stokes equation as well
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as Gallavotti's reversible Navier-Stokes equation and compare them.
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**Nstrophy is under active development**
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# Building
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Compile Nstrophy with
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```bash
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make
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```
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which will place a binary at `build/nstrophy`.
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The syntax for the execution of Nstrophy is
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```bash
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./build/nstrophy [-p parameters] [-s savefile] [-u u_outfile] [-t nthreads] <command>
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```
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* `parameters` is a list of parameters for the computation, see
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[Parameters](#parameters)
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* `savefile` is a file where the last step of the computation is saved in
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binary format so that the computation can be resumed after it has terminated,
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see
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[Interrupting and resuming the computation](#interrupting-and-resuming-the-computation).
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* `u_outfile` is a file to which the final u is written in plain text format,
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which can be used as an initial condition for a future computation.
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* `nthreads` is the number of threads used to compute Fast Fourier Transforms.
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Nstrophy is written in C. The Makefile uses the GNU C Compiler.
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Nstrophy depends on `fftw`: [https://fftw.org]
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# Commands
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The available commands are
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* `enstrophy`: to compute the enstrophy and various other observables. This
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command prints
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```
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step_index time average(alpha) average(energy) average(enstrophy) alpha energy enstrophy Re(u(1,1)) Re(u(1,2))
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```
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where the averages are running averages over `print_freq`. In addition, if
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the algorithm has an adaptive step, an extra column is printed with `delta`.
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In addition, if alpha has a negative value (even in between `print_freq`
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intervals), a line is printed with the information. The two components (1,1)
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and (1,2) of u are included to more easily identify periodic trajectories, or
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the presence of multiple attractors.
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* `lyapunov`: to compute the Lyapunov exponents. This command prints
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```
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time instantaneous_lyapunov lyapunov
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```
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where `instantaneous_lyapunov` is computed from the tangent flow only between
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the given time and the previous one.
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* `uk`: to compute the Fourier transform of the solution.
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* `quiet`: does not print anything (useful for debugging).
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* `enstrophy_print_init`: to compute the enstrophy and various other
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observables for the initial condition (useful for debugging). The command
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prints
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```
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alpha energy enstrophy
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```
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# Parameters
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The parameters can be specified using the `-p` flag. The parameter string
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should be a `;` sperated list of `key=value` pairs. The possible keys are
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* `equation`: either `irreverisible` (default) or `reversible`.
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* `K` (int, default 16): cutoff in momentum space: -K<=k_i<=K
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* `K1` (int, default `K`): cutoff in momentum space for the x component:
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-K<=k_x<=K
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* `K2` (int, default `K`): cutoff in momentum space for the y component:
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-K<=k_y<=K
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* `N` (int, default smallest power of 2 that is larger than 3`K`): size of fft
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vectors: must be at least 3 times `K` to avoid aliasing.
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* `N1` (int, default `N`): same as `N` but only for x component.
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* `N2` (int, default `N`): same as `N` but only for y component.
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* `final_time` (double, default 100000): time at which to end the computation.
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Set to <0 to keep on going forever.
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* `nu` (double, default 0.00048828125): viscosity.
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* `delta` (double, default 0.0001220703125): step size.
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* `L` (double, default 2pi): size of box.
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* `print_freq` (double, default 1): only print when time crosses integer
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multiples of `print_freq`.
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* `starting_time` (double, default 0): starting time.
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* `driving`: either `zero` for no driving, `test` (default) for a testing
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driving force or `file:<filename>` or `file_txt:<filename>` to read the
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driving force from a file. When using `file:<filename>` the file should be
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binary, whereas with `file_txt:<filename>` it should be plaintext. The binary
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file format is `(double)(double)` for each entry of the driving force,
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excluding kx<0 and kx=0&&ky<=0. The plaintext file format is
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`kx ky real_part imag_part`.
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* `init`: either `random` for a random initialization, `gaussian` (default) for
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a Gaussian initial condition or `file:<filename>` or `file_txt:<filename>` to
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read the driving force from a file. When using `file:<filename>` the file
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should be binary, whereas with `file_txt:<filename>` it should be plaintext.
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The binary file format is `(double)(double)` for each entry of the driving
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force, excluding kx<0 and kx=0&&ky<=0. The plaintext file format is
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`kx ky real_part imag_part`.
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* `init_energy` (double, default is to not rescale the initial condition, is
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incompatible with `init_enstrophy`): enforce a value for the energy of the
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initial condition.
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* `init_enstrophy` (double, default is to not rescale the initial condition, is
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incompatible with `init_energy`): enforce a value for the enstrophy of the
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initial condition.
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* `random_seed` (int, default ): seed for random initialization.
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* `algorithm`: fixed step methods: `RK4` for Runge-Kutta 4, `RK2` for
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Runge-Kutta 2.
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adaptive step methods: `RKF45` for Runge-Kutta-Fehlberg (order
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4), `RKDP54` for Runge-Kutta-Dormand-Prince (order 5), `RKBS32` for
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Runge-Kutta-Bogacki-Shampine (order 3).
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* `adaptive_tolerance` (double, default 1e-11): when using an adaptive step
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method, this is the maximal allowed relative error.
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* `adaptive_factor` (double, default 0.9): when using the RKF45 method, the
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step gets adjusted by `factor*delta*(tolerance/error)^(1/5)`.
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* `max_delta` (double, default 1e-2): when using the adaptive step, do not
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exceet `delta_max`.
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* `adaptive_cost`: cost function to use to estimate the error of the adaptive
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method: `L1` (default) for the normalized L1 norm, `k3` for the normalized L1
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norm of f_k/|k|^3, `k32` for the normalized L1 norm, `enstrophy` for the
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enstrophy, `alpha` for alpha.
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* `keep_en_cst` (0 or 1, default 0): impose that the enstrophy is constant at
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each step (only really useful for the reversible equation).
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* `print_alpha` (0 or 1, default 0): if this is set to 1, then whenever alpha
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is negative, its value is printed as a comment.
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* `lyapunov_reset` (double, default: `print_freq`): if this is set, then, when
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computing the Lyapunov exponents, the tangent flow will renormalize itself at
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times proportional to `lyapunov_reset`. This option is incompatible with
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`lyapunov_maxu`.
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* `lyapunov_maxu` (double, default: unset): if this is set, then, when
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computing the Lyapunov exponents, the tangent flow will renormalize itself
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whenever the norm of the tangent flow exceeds `lyapunov_maxu`. This option
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is incompatible with `lyapunov_reset`.
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* `algorithm_lyapunov`: the algorithm used to integrate the tangent flow. Can
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be `RK4` for Runge-Kutta 4 (default) or `RK2` for Runge-Kutta 2. Adaptive
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step algorithms cannot be used for the tangent flow.
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# Interrupting and resuming the computation
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The `enstrophy` and `lyapunov` computations can be interrupted by sending
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Nstrophy the `SIGINT` signal (e.g. by pressing `Ctrl-C`.) When Nstrophy
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receives the `SIGINT` signal, it finishes its current step and writes the value
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of uk, either to `savefile` if such a file was specified on the command line
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(using the `-s` flag), or to `stderr`. In addition, when a `savefile` is
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specified it writes the command that needs to be used to resume the computation
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(which essentially just sets the appropriate `starting_time` and
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`init:file:<savefile>` parameters). The data written to the `savefile` is
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binary.
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# License
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Nstrophy is released under the Apache 2.0 license.
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Copyright 2017-2025 Ian Jauslin
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