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Ian Jauslin 0244f03d02 Update copyright

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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

make

which will place a binary at build/nstrophy.

The syntax for the execution of Nstrophy is

./build/nstrophy [-p parameters] [-s savefile] [-u u_outfile] [-t nthreads] <command>

parameters is a list of parameters for the computation, see 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.
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 3K): 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.