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.

• 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.

Interrupting and resuming the computation

The 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