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Nstrophy
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keep_en_cst

This commit is contained in:
Ian Jauslin 2023-06-29 16:58:33 -04:00
parent 9c3ad2c400
commit 60e04ac681
4 changed files with 111 additions and 31 deletions
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3
README.md
View File

@ -125,6 +125,9 @@ should be a `;` sperated list of `key=value` pairs. The possible keys are
f_k/|k|^3, `k32` for the normalized L1 norm, `enstrophy` for the enstrophy
norm.
* `keep_en_cst` (0 or 1, default 0): impose that the enstrophy is constant at
each step (only really useful for the reversible equation).
# Interrupting and resuming the computation

17
src/main.c
View File

@ -50,6 +50,7 @@ typedef struct nstrophy_parameters {
unsigned int driving;
unsigned int init;
unsigned int algorithm;
bool keep_en_cst;
FILE* initfile;
FILE* drivingfile;
} nstrophy_parameters;
@ -178,16 +179,16 @@ int main (
// run command
if (command==COMMAND_UK){
uk(parameters.K1, parameters.K2, parameters.N1, parameters.N2, parameters.final_time, parameters.nu, parameters.delta, parameters.L, parameters.adaptive_tolerance, parameters.adaptive_factor, parameters.max_delta, parameters.adaptive_norm, u0, g, parameters.irreversible, parameters.algorithm, parameters.print_freq, parameters.starting_time, nthreads, savefile);
uk(parameters.K1, parameters.K2, parameters.N1, parameters.N2, parameters.final_time, parameters.nu, parameters.delta, parameters.L, parameters.adaptive_tolerance, parameters.adaptive_factor, parameters.max_delta, parameters.adaptive_norm, u0, g, parameters.irreversible, parameters.keep_en_cst, parameters.init_en, parameters.algorithm, parameters.print_freq, parameters.starting_time, nthreads, savefile);
}
else if(command==COMMAND_ENSTROPHY){
// register signal handler to handle aborts
signal(SIGINT, sig_handler);
signal(SIGTERM, sig_handler);
enstrophy(parameters.K1, parameters.K2, parameters.N1, parameters.N2, parameters.final_time, parameters.nu, parameters.delta, parameters.L, parameters.adaptive_tolerance, parameters.adaptive_factor, parameters.max_delta, parameters.adaptive_norm, u0, g, parameters.irreversible, parameters.algorithm, parameters.print_freq, parameters.starting_time, nthreads, savefile, utfile, (char*)argv[0], param_str, savefile_str, utfile_str);
enstrophy(parameters.K1, parameters.K2, parameters.N1, parameters.N2, parameters.final_time, parameters.nu, parameters.delta, parameters.L, parameters.adaptive_tolerance, parameters.adaptive_factor, parameters.max_delta, parameters.adaptive_norm, u0, g, parameters.irreversible, parameters.keep_en_cst, parameters.init_en, parameters.algorithm, parameters.print_freq, parameters.starting_time, nthreads, savefile, utfile, (char*)argv[0], param_str, savefile_str, utfile_str);
}
else if(command==COMMAND_QUIET){
quiet(parameters.K1, parameters.K2, parameters.N1, parameters.N2, parameters.final_time, parameters.nu, parameters.delta, parameters.L, parameters.adaptive_tolerance, parameters.adaptive_factor, parameters.max_delta, parameters.adaptive_norm, parameters.starting_time, u0, g, parameters.irreversible, parameters.algorithm, nthreads, savefile);
quiet(parameters.K1, parameters.K2, parameters.N1, parameters.N2, parameters.final_time, parameters.nu, parameters.delta, parameters.L, parameters.adaptive_tolerance, parameters.adaptive_factor, parameters.max_delta, parameters.adaptive_norm, parameters.starting_time, u0, g, parameters.irreversible, parameters.keep_en_cst, parameters.init_en, parameters.algorithm, nthreads, savefile);
}
else if(command==0){
fprintf(stderr, "error: no command specified\n");
@ -448,6 +449,7 @@ int read_params(
parameters->init=INIT_GAUSSIAN;
parameters->initfile=NULL;
parameters->algorithm=ALGORITHM_RK4;
parameters->keep_en_cst=false;
if (param_str!=NULL){
// init
@ -756,6 +758,15 @@ int set_parameter(
return(-1);
}
}
else if (strcmp(lhs,"keep_en_cst")==0){
int tmp;
ret=sscanf(rhs,"%d",&tmp);
if(ret!=1 || (tmp!=0 && tmp!=1)){
fprintf(stderr, "error: parameter 'keep_en_cst' should be 0 or 1\n got '%s'\n",rhs);
return(-1);
}
parameters->keep_en_cst=(tmp==1);
}
else{
fprintf(stderr, "error: unrecognized parameter '%s'\n",lhs);
return(-1);

106
src/navier-stokes.c
View File

@ -40,6 +40,8 @@ int uk(
_Complex double* u0,
_Complex double* g,
bool irreversible,
bool keep_en_cst,
double target_en,
unsigned int algorithm,
double print_freq,
double starting_time,
@ -88,17 +90,17 @@ int uk(
time=starting_time;
while(final_time<0 || time<final_time){
if(algorithm==ALGORITHM_RK2){
ns_step_rk2(u, K1, K2, N1, N2, nu, step, L, g, fft1, fft2, ifft, tmp1, tmp2, irreversible);
ns_step_rk2(u, K1, K2, N1, N2, nu, step, L, g, fft1, fft2, ifft, tmp1, tmp2, irreversible, keep_en_cst, target_en);
} else if (algorithm==ALGORITHM_RK4) {
ns_step_rk4(u, K1, K2, N1, N2, nu, step, L, g, fft1, fft2, ifft, tmp1, tmp2, tmp3, irreversible);
ns_step_rk4(u, K1, K2, N1, N2, nu, step, L, g, fft1, fft2, ifft, tmp1, tmp2, tmp3, irreversible, keep_en_cst, target_en);
} else if (algorithm==ALGORITHM_RKF45) {
ns_step_rkf45(u, adaptive_tolerance, adaptive_factor, max_delta, adaptive_norm, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, tmp1, tmp2, tmp3, tmp4, tmp5, tmp6, tmp7, irreversible, true);
ns_step_rkf45(u, adaptive_tolerance, adaptive_factor, max_delta, adaptive_norm, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, tmp1, tmp2, tmp3, tmp4, tmp5, tmp6, tmp7, irreversible, keep_en_cst, target_en, true);
} else if (algorithm==ALGORITHM_RKDP54) {
// only compute k1 on the first step
ns_step_rkdp54(u, adaptive_tolerance, adaptive_factor, max_delta, adaptive_norm, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, &tmp1, &tmp2, tmp3, tmp4, tmp5, tmp6, tmp7, irreversible, time==starting_time);
ns_step_rkdp54(u, adaptive_tolerance, adaptive_factor, max_delta, adaptive_norm, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, &tmp1, &tmp2, tmp3, tmp4, tmp5, tmp6, tmp7, irreversible, keep_en_cst, target_en, time==starting_time);
} else if (algorithm==ALGORITHM_RKBS32) {
// only compute k1 on the first step
ns_step_rkbs32(u, adaptive_tolerance, adaptive_factor, max_delta, adaptive_norm, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, &tmp1, tmp2, tmp3, &tmp4, tmp5, irreversible, time==starting_time);
ns_step_rkbs32(u, adaptive_tolerance, adaptive_factor, max_delta, adaptive_norm, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, &tmp1, tmp2, tmp3, &tmp4, tmp5, irreversible, keep_en_cst, target_en, time==starting_time);
} else {
fprintf(stderr,"bug: unknown algorithm: %u, contact ian.jauslin@rutgers,edu\n",algorithm);
}
@ -151,6 +153,8 @@ int enstrophy(
_Complex double* u0,
_Complex double* g,
bool irreversible,
bool keep_en_cst,
double target_en,
unsigned int algorithm,
double print_freq,
double starting_time,
@ -202,17 +206,17 @@ int enstrophy(
time=starting_time;
while(final_time<0 || time<final_time){
if(algorithm==ALGORITHM_RK2){
ns_step_rk2(u, K1, K2, N1, N2, nu, step, L, g, fft1, fft2, ifft, tmp1, tmp2, irreversible);
ns_step_rk2(u, K1, K2, N1, N2, nu, step, L, g, fft1, fft2, ifft, tmp1, tmp2, irreversible, keep_en_cst, target_en);
} else if (algorithm==ALGORITHM_RK4) {
ns_step_rk4(u, K1, K2, N1, N2, nu, step, L, g, fft1, fft2, ifft, tmp1, tmp2, tmp3, irreversible);
ns_step_rk4(u, K1, K2, N1, N2, nu, step, L, g, fft1, fft2, ifft, tmp1, tmp2, tmp3, irreversible, keep_en_cst, target_en);
} else if (algorithm==ALGORITHM_RKF45) {
ns_step_rkf45(u, adaptive_tolerance, adaptive_factor, max_delta, adaptive_norm, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, tmp1, tmp2, tmp3, tmp4, tmp5, tmp6, tmp7, irreversible, true);
ns_step_rkf45(u, adaptive_tolerance, adaptive_factor, max_delta, adaptive_norm, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, tmp1, tmp2, tmp3, tmp4, tmp5, tmp6, tmp7, irreversible, keep_en_cst, target_en, true);
} else if (algorithm==ALGORITHM_RKDP54) {
// only compute k1 on the first step
ns_step_rkdp54(u, adaptive_tolerance, adaptive_factor, max_delta, adaptive_norm, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, &tmp1, &tmp2, tmp3, tmp4, tmp5, tmp6, tmp7, irreversible, time==starting_time);
ns_step_rkdp54(u, adaptive_tolerance, adaptive_factor, max_delta, adaptive_norm, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, &tmp1, &tmp2, tmp3, tmp4, tmp5, tmp6, tmp7, irreversible, keep_en_cst, target_en, time==starting_time);
} else if (algorithm==ALGORITHM_RKBS32) {
// only compute k1 on the first step
ns_step_rkbs32(u, adaptive_tolerance, adaptive_factor, max_delta, adaptive_norm, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, &tmp1, tmp2, tmp3, &tmp4, tmp5, irreversible, time==starting_time);
ns_step_rkbs32(u, adaptive_tolerance, adaptive_factor, max_delta, adaptive_norm, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, &tmp1, tmp2, tmp3, &tmp4, tmp5, irreversible, keep_en_cst, target_en, time==starting_time);
} else {
fprintf(stderr,"bug: unknown algorithm: %u, contact ian.jauslin@rutgers,edu\n",algorithm);
}
@ -345,6 +349,8 @@ int quiet(
_Complex double* u0,
_Complex double* g,
bool irreversible,
bool keep_en_cst,
double target_en,
unsigned int algorithm,
unsigned int nthreads,
FILE* savefile
@ -375,17 +381,17 @@ int quiet(
time=starting_time;
while(final_time<0 || time<final_time){
if(algorithm==ALGORITHM_RK2){
ns_step_rk2(u, K1, K2, N1, N2, nu, step, L, g, fft1, fft2, ifft, tmp1, tmp2, irreversible);
ns_step_rk2(u, K1, K2, N1, N2, nu, step, L, g, fft1, fft2, ifft, tmp1, tmp2, irreversible, keep_en_cst, target_en);
} else if (algorithm==ALGORITHM_RK4) {
ns_step_rk4(u, K1, K2, N1, N2, nu, step, L, g, fft1, fft2, ifft, tmp1, tmp2, tmp3, irreversible);
ns_step_rk4(u, K1, K2, N1, N2, nu, step, L, g, fft1, fft2, ifft, tmp1, tmp2, tmp3, irreversible, keep_en_cst, target_en);
} else if (algorithm==ALGORITHM_RKF45) {
ns_step_rkf45(u, adaptive_tolerance, adaptive_factor, max_delta, adaptive_norm, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, tmp1, tmp2, tmp3, tmp4, tmp5, tmp6, tmp7, irreversible, true);
ns_step_rkf45(u, adaptive_tolerance, adaptive_factor, max_delta, adaptive_norm, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, tmp1, tmp2, tmp3, tmp4, tmp5, tmp6, tmp7, irreversible, keep_en_cst, target_en, true);
} else if (algorithm==ALGORITHM_RKDP54) {
// only compute k1 on the first step
ns_step_rkdp54(u, adaptive_tolerance, adaptive_factor, max_delta, adaptive_norm, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, &tmp1, &tmp2, tmp3, tmp4, tmp5, tmp6, tmp7, irreversible, time==starting_time);
ns_step_rkdp54(u, adaptive_tolerance, adaptive_factor, max_delta, adaptive_norm, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, &tmp1, &tmp2, tmp3, tmp4, tmp5, tmp6, tmp7, irreversible, keep_en_cst, target_en, time==starting_time);
} else if (algorithm==ALGORITHM_RKBS32) {
// only compute k1 on the first step
ns_step_rkbs32(u, adaptive_tolerance, adaptive_factor, max_delta, adaptive_norm, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, &tmp1, tmp2, tmp3, &tmp4, tmp5, irreversible, time==starting_time);
ns_step_rkbs32(u, adaptive_tolerance, adaptive_factor, max_delta, adaptive_norm, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, &tmp1, tmp2, tmp3, &tmp4, tmp5, irreversible, keep_en_cst, target_en, time==starting_time);
} else {
fprintf(stderr,"bug: unknown algorithm: %u, contact ian.jauslin@rutgers,edu\n",algorithm);
}
@ -558,7 +564,9 @@ int ns_step_rk4(
_Complex double* tmp1,
_Complex double* tmp2,
_Complex double* tmp3,
bool irreversible
bool irreversible,
bool keep_en_cst,
double target_en
){
int kx,ky;
@ -616,6 +624,16 @@ int ns_step_rk4(
}
}
// keep enstrophy constant
if(keep_en_cst){
double en=compute_enstrophy(u, K1, K2, L);
for(kx=0;kx<=K1;kx++){
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
u[klookup_sym(kx,ky,K2)]*=sqrt(target_en/en);
}
}
}
return(0);
}
@ -635,7 +653,9 @@ int ns_step_rk2(
fft_vect ifft,
_Complex double* tmp1,
_Complex double* tmp2,
bool irreversible
bool irreversible,
bool keep_en_cst,
double target_en
){
int kx,ky;
@ -657,6 +677,16 @@ int ns_step_rk2(
}
}
// keep enstrophy constant
if(keep_en_cst){
double en=compute_enstrophy(u, K1, K2, L);
for(kx=0;kx<=K1;kx++){
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
u[klookup_sym(kx,ky,K2)]*=sqrt(target_en/en);
}
}
}
return(0);
}
@ -688,6 +718,8 @@ int ns_step_rkf45(
_Complex double* k6,
_Complex double* tmp,
bool irreversible,
bool keep_en_cst,
double target_en,
// whether to compute k1 or leave it as is
bool compute_k1
){
@ -805,12 +837,22 @@ int ns_step_rkf45(
}
// next delta to use in future steps (do not exceed max_delta)
*next_delta=fmin(max_delta, (*delta)*pow(relative*tolerance/err,0.2));
// keep enstrophy constant
if(keep_en_cst){
double en=compute_enstrophy(u, K1, K2, L);
for(kx=0;kx<=K1;kx++){
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
u[klookup_sym(kx,ky,K2)]*=sqrt(target_en/en);
}
}
}
}
// error too big: repeat with smaller step
else{
*delta=factor*(*delta)*pow(relative*tolerance/err,0.2);
// do not recompute k1
ns_step_rkf45(u,tolerance,factor,max_delta,adaptive_norm,K1,K2,N1,N2,nu,delta,next_delta,L,g,fft1,fft2,ifft,k1,k2,k3,k4,k5,k6,tmp,irreversible,false);
ns_step_rkf45(u,tolerance,factor,max_delta,adaptive_norm,K1,K2,N1,N2,nu,delta,next_delta,L,g,fft1,fft2,ifft,k1,k2,k3,k4,k5,k6,tmp,irreversible,keep_en_cst,target_en,false);
}
return 0;
@ -844,6 +886,8 @@ int ns_step_rkbs32(
_Complex double** k4,
_Complex double* tmp,
bool irreversible,
bool keep_en_cst,
double target_en,
// whether to compute k1
bool compute_k1
){
@ -944,12 +988,22 @@ int ns_step_rkbs32(
tmp=*k1;
*k1=*k4;
*k4=tmp;
// keep enstrophy constant
if(keep_en_cst){
double en=compute_enstrophy(u, K1, K2, L);
for(kx=0;kx<=K1;kx++){
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
u[klookup_sym(kx,ky,K2)]*=sqrt(target_en/en);
}
}
}
}
// error too big: repeat with smaller step
else{
*delta=factor*(*delta)*pow(relative*tolerance/err,1./3);
// this will reuse the same k1 without re-computing it
ns_step_rkbs32(u,tolerance,factor,max_delta,adaptive_norm,K1,K2,N1,N2,nu,delta,next_delta,L,g,fft1,fft2,ifft,k1,k2,k3,k4,tmp,irreversible,false);
ns_step_rkbs32(u,tolerance,factor,max_delta,adaptive_norm,K1,K2,N1,N2,nu,delta,next_delta,L,g,fft1,fft2,ifft,k1,k2,k3,k4,tmp,irreversible,keep_en_cst,target_en,false);
}
return 0;
@ -984,6 +1038,8 @@ int ns_step_rkdp54(
_Complex double* k6,
_Complex double* tmp,
bool irreversible,
bool keep_en_cst,
double target_en,
// whether to compute k1
bool compute_k1
){
@ -1109,12 +1165,22 @@ int ns_step_rkdp54(
tmp=*k1;
*k1=*k2;
*k2=tmp;
// keep enstrophy constant
if(keep_en_cst){
double en=compute_enstrophy(u, K1, K2, L);
for(kx=0;kx<=K1;kx++){
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
u[klookup_sym(kx,ky,K2)]*=sqrt(target_en/en);
}
}
}
}
// error too big: repeat with smaller step
else{
*delta=factor*(*delta)*pow(relative*tolerance/err,1./5);
// this will reuse the same k1 without re-computing it
ns_step_rkdp54(u,tolerance,factor,max_delta,adaptive_norm,K1,K2,N1,N2,nu,delta,next_delta,L,g,fft1,fft2,ifft,k1,k2,k3,k4,k5,k6,tmp,irreversible,false);
ns_step_rkdp54(u,tolerance,factor,max_delta,adaptive_norm,K1,K2,N1,N2,nu,delta,next_delta,L,g,fft1,fft2,ifft,k1,k2,k3,k4,k5,k6,tmp,irreversible,keep_en_cst,target_en,false);
}
return 0;

16
src/navier-stokes.h
View File

@ -33,13 +33,13 @@ typedef struct fft_vects {
} fft_vect;
// compute u_k
int uk( int K1, int K2, int N1, int N2, double final_time, double nu, double delta, double L, double adaptive_tolerance, double adaptive_factor, double max_delta, unsigned int adaptive_norm, _Complex double* u0, _Complex double* g, bool irreversible, unsigned int algorithm, double print_freq, double starting_time, unsigned int nthreadsl, FILE* savefile);
int uk( int K1, int K2, int N1, int N2, double final_time, double nu, double delta, double L, double adaptive_tolerance, double adaptive_factor, double max_delta, unsigned int adaptive_norm, _Complex double* u0, _Complex double* g, bool irreversible, bool keep_en_cst, double target_en, unsigned int algorithm, double print_freq, double starting_time, unsigned int nthreadsl, FILE* savefile);
// compute enstrophy and alpha
int enstrophy( int K1, int K2, int N1, int N2, double final_time, double nu, double delta, double L, double adaptive_tolerance, double adaptive_factor, double max_delta, unsigned int adaptive_norm, _Complex double* u0, _Complex double* g, bool irreversible, unsigned int algorithm, double print_freq, double starting_time, unsigned int nthreads, FILE* savefile, FILE* utfile, char* cmd_string, char* params_string, char* savefile_string, char* utfile_string);
int enstrophy( int K1, int K2, int N1, int N2, double final_time, double nu, double delta, double L, double adaptive_tolerance, double adaptive_factor, double max_delta, unsigned int adaptive_norm, _Complex double* u0, _Complex double* g, bool irreversible, bool keep_en_cst, double target_en, unsigned int algorithm, double print_freq, double starting_time, unsigned int nthreads, FILE* savefile, FILE* utfile, char* cmd_string, char* params_string, char* savefile_string, char* utfile_string);
// compute solution as a function of time, but do not print anything (useful for debugging)
int quiet( int K1, int K2, int N1, int N2, double final_time, double nu, double delta, double L, double adaptive_tolerance, double adaptive_factor, double max_delta, unsigned int adaptive_norm, double starting_time, _Complex double* u0, _Complex double* g, bool irreversible, unsigned int algorithm, unsigned int nthreads, FILE* savefile);
int quiet( int K1, int K2, int N1, int N2, double final_time, double nu, double delta, double L, double adaptive_tolerance, double adaptive_factor, double max_delta, unsigned int adaptive_norm, double starting_time, _Complex double* u0, _Complex double* g, bool irreversible, bool keep_en_cst, double target_en, unsigned int algorithm, unsigned int nthreads, FILE* savefile);
// initialize vectors for computation
@ -52,15 +52,15 @@ int copy_u( _Complex double* u, _Complex double* u0, int K1, int K2);
// next time step for Irreversible/reversible Navier-Stokes equation
// RK4
int ns_step_rk4( _Complex double* u, int K1, int K2, int N1, int N2, double nu, double delta, double L, _Complex double* g, fft_vect fft1, fft_vect fft2,fft_vect ifft, _Complex double* tmp1, _Complex double *tmp2, _Complex double *tmp3, bool irreversible);
int ns_step_rk4( _Complex double* u, int K1, int K2, int N1, int N2, double nu, double delta, double L, _Complex double* g, fft_vect fft1, fft_vect fft2,fft_vect ifft, _Complex double* tmp1, _Complex double *tmp2, _Complex double *tmp3, bool irreversible, bool keep_en_cst, double target_en);
// RK2
int ns_step_rk2( _Complex double* u, int K1, int K2, int N1, int N2, double nu, double delta, double L, _Complex double* g, fft_vect fft1, fft_vect fft2,fft_vect ifft, _Complex double* tmp1, _Complex double *tmp2, bool irreversible);
int ns_step_rk2( _Complex double* u, int K1, int K2, int N1, int N2, double nu, double delta, double L, _Complex double* g, fft_vect fft1, fft_vect fft2,fft_vect ifft, _Complex double* tmp1, _Complex double *tmp2, bool irreversible, bool keep_en_cst, double target_en);
// Runge-Kutta-Fehlberg
int ns_step_rkf45( _Complex double* u, double tolerance, double factor, double max_delta, unsigned int adaptive_norm, int K1, int K2, int N1, int N2, double nu, double* delta, double* next_delta, double L, _Complex double* g, fft_vect fft1, fft_vect fft2, fft_vect ifft, _Complex double* k1, _Complex double* k2, _Complex double* k3, _Complex double* k4, _Complex double* k5, _Complex double* k6, _Complex double* tmp, bool irreversible, bool compute_k1);
int ns_step_rkf45( _Complex double* u, double tolerance, double factor, double max_delta, unsigned int adaptive_norm, int K1, int K2, int N1, int N2, double nu, double* delta, double* next_delta, double L, _Complex double* g, fft_vect fft1, fft_vect fft2, fft_vect ifft, _Complex double* k1, _Complex double* k2, _Complex double* k3, _Complex double* k4, _Complex double* k5, _Complex double* k6, _Complex double* tmp, bool irreversible, bool keep_en_cst, double target_en, bool compute_k1);
// Runge-Kutta-Dromand-Prince
int ns_step_rkdp54( _Complex double* u, double tolerance, double factor, double max_delta, unsigned int adaptive_norm, int K1, int K2, int N1, int N2, double nu, double* delta, double* next_delta, double L, _Complex double* g, fft_vect fft1, fft_vect fft2, fft_vect ifft, _Complex double** k1, _Complex double** k2, _Complex double* k3, _Complex double* k4, _Complex double* k5, _Complex double* k6, _Complex double* tmp, bool irreversible, bool compute_k1);
int ns_step_rkdp54( _Complex double* u, double tolerance, double factor, double max_delta, unsigned int adaptive_norm, int K1, int K2, int N1, int N2, double nu, double* delta, double* next_delta, double L, _Complex double* g, fft_vect fft1, fft_vect fft2, fft_vect ifft, _Complex double** k1, _Complex double** k2, _Complex double* k3, _Complex double* k4, _Complex double* k5, _Complex double* k6, _Complex double* tmp, bool irreversible, bool keep_en_cst, double target_en, bool compute_k1);
// Runge-Kutta-Bogacki-Shampine
int ns_step_rkbs32( _Complex double* u, double tolerance, double factor, double max_delta, unsigned int adaptive_norm, int K1, int K2, int N1, int N2, double nu, double* delta, double* next_delta, double L, _Complex double* g, fft_vect fft1, fft_vect fft2, fft_vect ifft, _Complex double** k1, _Complex double* k2, _Complex double* k3, _Complex double** k4, _Complex double* tmp, bool irreversible, bool compute_k1);
int ns_step_rkbs32( _Complex double* u, double tolerance, double factor, double max_delta, unsigned int adaptive_norm, int K1, int K2, int N1, int N2, double nu, double* delta, double* next_delta, double L, _Complex double* g, fft_vect fft1, fft_vect fft2, fft_vect ifft, _Complex double** k1, _Complex double* k2, _Complex double* k3, _Complex double** k4, _Complex double* tmp, bool irreversible, bool keep_en_cst, double target_en, bool compute_k1);
// right side of Irreversible/reversible Navier-Stokes equation
int ns_rhs( _Complex double* out, _Complex double* u, int K1, int K2, int N1, int N2, double nu, double L, _Complex double* g, fft_vect fft1, fft_vect fft2, fft_vect ifft, bool irreversible);