delta_max and fix cabs2

README.md

@@ -105,6 +105,9 @@ should be a `;` sperated list of `key=value` pairs. The possible keys are
 * `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`.
+
 
 # Interrupting/resuming the computation
 

docs/nstrophy_doc.tex

@@ -103,6 +103,88 @@ We truncate the Fourier modes and assume that $\hat u_k=0$ if $|k_1|>K_1$ or $|k
 \end{equation}
 \bigskip
 
+\point{\bf Runge-Kutta methods}.
+To solve the equation numerically, we will use Runge-Kutta methods, which compute an approximate value $\hat u_k^{(n)}$ for $\hat u_k(t_n)$.
+{\tt nstrophy} supports the 4th order Runge-Kutta ({\tt RK4}) and 2nd order Runge-Kutta ({\tt RK2}) algorithms.
+In addition, several variable step methods are implemented:
+\begin{itemize}
+  \item the Runge-Kutta-Dormand-Prince method ({\tt RKDP54}), which is of 5th order, and adjusts the step by comparing to a 4th order method;
+  \item the Runge-Kutta-Fehlberg method ({\tt RKF45}), which is of 4th order, and adjusts the step by comparing to a 5th order method;
+  \item the Runge-Kutta-Bogacki-Shampine method ({\tt RKBS32}), which is of 3d order, and adjusts the step by comparing to a 2nd order method.
+\end{itemize}
+In these adaptive step methods, two steps are computed at different orders: $\hat u_k^{(n)}$ and $\hat U_k^{(n)}$, the step size is adjusted at every step in such a way that the error is small enough:
+\begin{equation}
+  \|\hat u^{(n)}-\hat U^{(n)}\|
+  <\epsilon_{\mathrm{target}}
+\end{equation}
+for some given $\epsilon_{\mathrm{target}}$, set using the {\tt adaptive\_tolerance} parameter.
+The choice of the norm matters, and will be discussed below.
+If the error is larger than the target, then the step size is decreased.
+How this is done depends on the order of algorithm.
+If the order is $q$ (here we mean the smaller of the two orders, so 4 for {\tt RKDP54} and {\tt RKF45} and 2 for {\tt RKBS32}), then we expect
+\begin{equation}
+  \|\hat u^{(n)}-\hat U^{(n)}\|=\delta_n^qC_n
+  .
+\end{equation}
+We wish to set $\delta_{n+1}$ so that
+\begin{equation}
+  \delta_{n+1}^qC_n=\epsilon_{\mathrm{target}}
+\end{equation}
+so
+\begin{equation}
+  \delta_{n+1}
+  =\left(\frac{\epsilon_{\mathrm{target}}}{C_n}\right)^{\frac1q}
+  =\delta_n\left(\frac{\epsilon_{\mathrm{target}}}{\|\hat u^{(n)}-\hat U^{(n)}\|}\right)^{\frac1q}
+  .
+  \label{adaptive_delta}
+\end{equation}
+(Actually, to be safe and ensure that $\delta$ decreases sufficiently, we multiply this by a safety factor that can be set using the {\tt adaptive\_factor} parameter.)
+If the error is smaller than the target, we increase $\delta$ using\-~(\ref{adaptive_delta}) (without the safety factor).
+To be safe, we also set a maximal value for $\delta$ via the {\tt max\_delta} parameter.
+\bigskip
+
+\indent
+The choice of the norm $\|\cdot\|$ matters.
+\begin{itemize}
+  \item A naive choice is to take $\|\cdot\|$ to be the normalized $L_1$ norm:
+  \begin{equation}
+    \|f\|:=
+    \frac1{\mathcal N}\sum_k|f_k|
+    ,\quad
+    \mathcal N:=\sum_k|\hat u_k^{(n)}-\hat u_k^{(n-1)}|
+    .
+  \end{equation}
+
+  \item Empirically, we have found that $|\hat u-\hat U|$ behaves like $k^{-3}$ for {\tt RKDP54} and {\tt RKF45}, and like $k^{-\frac32}$ for {\tt RKBS32}, so a norm of the form
+  \begin{equation}
+    \|f\|:=\frac1{\mathcal N}\sum_k|f_k|k^{-3}
+    ,\quad
+    \mathcal N:=\sum_k|\hat u_k^{(n)}-\hat u_k^{(n-1)}|k^{-3}
+  \end{equation}
+  or
+  \begin{equation}
+    \|f\|:=\frac1{\mathcal N}\sum_k|f_k|k^{-\frac32}
+    ,\quad
+    \mathcal N:=\sum_k|\hat u_k^{(n)}-\hat u_k^{(n-1)}|k^{-\frac32}
+  \end{equation}
+  are sensible choices.
+
+  \item
+  Another option is to define a norm based on the expression of the enstrophy\-~(\ref{enstrophy}):
+  \begin{equation}
+    \|f\|:=\frac1{\mathcal N}\sqrt{\sum_k k^2|f_k|^2}
+    \quad
+    \mathcal N:=\sqrt{\sum_k k^2|\hat u_k^{(n)}|^2}
+    .
+  \end{equation}
+  Doing so controls the error of the enstrophy through
+  \begin{equation}
+    \mathcal N^2|\mathcal En(\hat u)-\mathcal En(\hat U)|\equiv|\|\hat u\|^2-\|\hat U\|^2|\leqslant \|\hat u-\hat U\|(\|\hat u\|+\|\hat U\|)
+    .
+  \end{equation}
+\end{itemize}
+\bigskip
+
 \point{\bf Reality}.
 Since $U$ is real, $\hat U_{-k}=\hat U_k^*$, and so
 \begin{equation}
@@ -271,6 +353,7 @@ and
     +e^{-\frac{4\pi^2}{L^2}\nu t}\sqrt{E(0)}
   \right)^2
   .
+  \label{enstrophy}
 \end{equation}
 \bigskip
 

src/complex_tools.c

@@ -0,0 +1,22 @@
+/*
+Copyright 2017-2023 Ian Jauslin
+
+Licensed under the Apache License, Version 2.0 (the "License");
+you may not use this file except in compliance with the License.
+You may obtain a copy of the License at
+
+    http://www.apache.org/licenses/LICENSE-2.0
+
+Unless required by applicable law or agreed to in writing, software
+distributed under the License is distributed on an "AS IS" BASIS,
+WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+See the License for the specific language governing permissions and
+limitations under the License.
+*/
+
+#include "complex_tools.h"
+
+// magnitude squared
+double cabs2(_Complex double x){
+  return (__real__ x)*(__real__ x)+(__imag__ x)*(__imag__ x);
+}

src/complex_tools.h

@@ -0,0 +1,23 @@
+/*
+Copyright 2017-2023 Ian Jauslin
+
+Licensed under the Apache License, Version 2.0 (the "License");
+you may not use this file except in compliance with the License.
+You may obtain a copy of the License at
+
+    http://www.apache.org/licenses/LICENSE-2.0
+
+Unless required by applicable law or agreed to in writing, software
+distributed under the License is distributed on an "AS IS" BASIS,
+WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+See the License for the specific language governing permissions and
+limitations under the License.
+*/
+
+#ifndef COMPLEXTOOLS_H
+#define COMPLEXTOOLS_H
+
+// magnitude squared
+double cabs2(_Complex double x);
+
+#endif

src/main.c

@@ -47,6 +47,7 @@ typedef struct nstrophy_parameters {
   double L;
   double adaptive_tolerance;
   double adaptive_factor;
+  double max_delta;
   double print_freq;
   int seed;
   double starting_time;
@@ -170,16 +171,16 @@ 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, 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, u0, g, parameters.irreversible, 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, u0, g, parameters.irreversible, parameters.algorithm, parameters.print_freq, parameters.starting_time, nthreads, savefile, (char*)argv[0], param_str, savefile_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, u0, g, parameters.irreversible, parameters.algorithm, parameters.print_freq, parameters.starting_time, nthreads, savefile, (char*)argv[0], param_str, savefile_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.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.starting_time, u0, g, parameters.irreversible, parameters.algorithm, nthreads, savefile);
   }
   else if(command==0){
     fprintf(stderr, "error: no command specified\n");
@@ -264,13 +265,13 @@ int print_params(
     fprintf(file,", algorithm=RK2");
     break;
   case ALGORITHM_RKF45:
-    fprintf(file,", algorithm=RKF45, tolerance=%.15e, factor=%.15e",parameters.adaptive_tolerance, parameters.adaptive_factor);
+    fprintf(file,", algorithm=RKF45, tolerance=%.15e",parameters.adaptive_tolerance);
     break;
   case ALGORITHM_RKDP54:
-    fprintf(file,", algorithm=RKDP54, tolerance=%.15e, factor=%.15e",parameters.adaptive_tolerance, parameters.adaptive_factor);
+    fprintf(file,", algorithm=RKDP54, tolerance=%.15e",parameters.adaptive_tolerance);
     break;
   case ALGORITHM_RKBS32:
-    fprintf(file,", algorithm=RKBS32, tolerance=%.15e, factor=%.15e",parameters.adaptive_tolerance, parameters.adaptive_factor);
+    fprintf(file,", algorithm=RKBS32, tolerance=%.15e",parameters.adaptive_tolerance);
     break;
   default:
     fprintf(file,", algorithm=RK4");
@@ -397,6 +398,7 @@ int read_params(
   parameters->L=2*M_PI;
   parameters->adaptive_tolerance=1e-11;
   parameters->adaptive_factor=0.9;
+  parameters->max_delta=1e-2;
   parameters->final_time=100000;
   parameters->print_freq=1;
   parameters->starting_time=0;
@@ -597,6 +599,13 @@ int set_parameter(
       return(-1);
     }
   }
+  else if (strcmp(lhs,"max_delta")==0){
+    ret=sscanf(rhs,"%lf",&(parameters->max_delta));
+    if(ret!=1){
+      fprintf(stderr, "error: parameter 'max_delta' should be a double\n       got '%s'\n",rhs);
+      return(-1);
+    }
+  }
   else if (strcmp(lhs,"print_freq")==0){
     ret=sscanf(rhs,"%lf",&(parameters->print_freq));
     if(ret!=1){

src/navier-stokes.c

@@ -17,13 +17,12 @@ limitations under the License.
 #include "constants.cpp"
 #include "io.h"
 #include "navier-stokes.h"
+#include "complex_tools.h"
 #include <math.h>
 #include <stdint.h>
 #include <stdlib.h>
 #include <string.h>
 
-#define CABS2(x) ((__real__ x) * (__real__ x) + (__imag__ x) * (__imag__ x))
-
 // compute solution as a function of time
 int uk(
   int K1,
@@ -36,6 +35,7 @@ int uk(
   double L,
   double adaptive_tolerance,
   double adaptive_factor,
+  double max_delta,
   _Complex double* u0,
   _Complex double* g,
   bool irreversible,
@@ -94,7 +94,7 @@ int uk(
       ns_step_rkf45(u, adaptive_tolerance, adaptive_factor, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, tmp1, tmp2, tmp3, tmp4, tmp5, tmp6, tmp7, irreversible, true);
     } else if (algorithm==ALGORITHM_RKDP54) {
       // only compute k1 on the first step
-      ns_step_rkdp54(u, adaptive_tolerance, adaptive_factor, 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, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, &tmp1, &tmp2, tmp3, tmp4, tmp5, tmp6, tmp7, irreversible, time==starting_time);
     } else if (algorithm==ALGORITHM_RKBS32) {
       // only compute k1 on the first step
       ns_step_rkbs32(u, adaptive_tolerance, adaptive_factor, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, &tmp1, tmp2, tmp3, &tmp4, tmp5, irreversible, time==starting_time);
@@ -144,6 +144,7 @@ int enstrophy(
   double L,
   double adaptive_tolerance,
   double adaptive_factor,
+  double max_delta,
   _Complex double* u0,
   _Complex double* g,
   bool irreversible,
@@ -203,7 +204,7 @@ int enstrophy(
       ns_step_rkf45(u, adaptive_tolerance, adaptive_factor, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, tmp1, tmp2, tmp3, tmp4, tmp5, tmp6, tmp7, irreversible, true);
     } else if (algorithm==ALGORITHM_RKDP54) {
       // only compute k1 on the first step
-      ns_step_rkdp54(u, adaptive_tolerance, adaptive_factor, 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, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, &tmp1, &tmp2, tmp3, tmp4, tmp5, tmp6, tmp7, irreversible, time==starting_time);
     } else if (algorithm==ALGORITHM_RKBS32) {
       // only compute k1 on the first step
       ns_step_rkbs32(u, adaptive_tolerance, adaptive_factor, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, &tmp1, tmp2, tmp3, &tmp4, tmp5, irreversible, time==starting_time);
@@ -317,6 +318,7 @@ int quiet(
   double final_time,
   double nu,
   double delta,
+  double max_delta,
   double L,
   double adaptive_tolerance,
   double adaptive_factor,
@@ -361,7 +363,7 @@ int quiet(
       ns_step_rkf45(u, adaptive_tolerance, adaptive_factor, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, tmp1, tmp2, tmp3, tmp4, tmp5, tmp6, tmp7, irreversible, true);
     } else if (algorithm==ALGORITHM_RKDP54) {
       // only compute k1 on the first step
-      ns_step_rkdp54(u, adaptive_tolerance, adaptive_factor, 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, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, &tmp1, &tmp2, tmp3, tmp4, tmp5, tmp6, tmp7, irreversible, time==starting_time);
     } else if (algorithm==ALGORITHM_RKBS32) {
       // only compute k1 on the first step
       ns_step_rkbs32(u, adaptive_tolerance, adaptive_factor, K1, K2, N1, N2, nu, &step, &next_step, L, g, fft1, fft2, ifft, &tmp1, tmp2, tmp3, &tmp4, tmp5, irreversible, time==starting_time);
@@ -856,6 +858,7 @@ int ns_step_rkdp54(
   _Complex double* u,
   double tolerance,
   double factor,
+  double max_delta,
   int K1,
   int K2,
   int N1,
@@ -946,10 +949,13 @@ int ns_step_rkdp54(
     for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
       // difference between 5th order and 4th order
       // use the norm |u_k|^2k^2 (to get a bound on the error of the enstrophy)
-      err+=(kx*kx+ky*ky)*CABS2((*delta)*(-71./57600*(*k1)[klookup_sym(kx,ky,K2)]+71./16695*k3[klookup_sym(kx,ky,K2)]-71./1920*k4[klookup_sym(kx,ky,K2)]+17253./339200*k5[klookup_sym(kx,ky,K2)]-22./525*k6[klookup_sym(kx,ky,K2)]+1./40*(*k2)[klookup_sym(kx,ky,K2)]));
+      err+=(kx*kx+ky*ky)*cabs2((*delta)*(-71./57600*(*k1)[klookup_sym(kx,ky,K2)]+71./16695*k3[klookup_sym(kx,ky,K2)]-71./1920*k4[klookup_sym(kx,ky,K2)]+17253./339200*k5[klookup_sym(kx,ky,K2)]-22./525*k6[klookup_sym(kx,ky,K2)]+1./40*(*k2)[klookup_sym(kx,ky,K2)]));
-      relative+=(kx*kx+ky*ky)*(CABS2(tmp[klookup_sym(kx,ky,K2)]-u[klookup_sym(kx,ky,K2)]));
+      //relative+=(kx*kx+ky*ky)*(CABS2(tmp[klookup_sym(kx,ky,K2)]-u[klookup_sym(kx,ky,K2)]));
+      relative+=(kx*kx+ky*ky)*(cabs2(u[klookup_sym(kx,ky,K2)]));
     }
   }
+  err=sqrt(err);
+  relative=sqrt(relative);
 
   // compare relative error with tolerance
   if(err<relative*tolerance){
@@ -959,8 +965,8 @@ int ns_step_rkdp54(
 	u[klookup_sym(kx,ky,K2)]=tmp[klookup_sym(kx,ky,K2)];
       }
     }
-    // next delta to use in future steps
+    // next delta to use in future steps (do not exceed max_delta)
-    *next_delta=(*delta)*pow(relative*tolerance/err,1./5);
+    *next_delta=fmin(max_delta, (*delta)*pow(relative*tolerance/err,1./5));
 
     // k1 in the next step will be this k4 (first same as last)
     tmp=*k1;
@@ -971,7 +977,7 @@ int ns_step_rkdp54(
   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,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,K1,K2,N1,N2,nu,delta,next_delta,L,g,fft1,fft2,ifft,k1,k2,k3,k4,k5,k6,tmp,irreversible,false);
   }
 
   return 0;

src/navier-stokes.h

@@ -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, _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, _Complex double* u0, _Complex double* g, bool irreversible, 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, _Complex double* u0, _Complex double* g, bool irreversible, unsigned int algorithm, double print_freq, double starting_time, unsigned int nthreads, FILE* savefile, char* cmd_string, char* params_string, char* savefile_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, _Complex double* u0, _Complex double* g, bool irreversible, unsigned int algorithm, double print_freq, double starting_time, unsigned int nthreads, FILE* savefile, char* cmd_string, char* params_string, char* savefile_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 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, double starting_time, _Complex double* u0, _Complex double* g, bool irreversible, unsigned int algorithm, unsigned int nthreads, FILE* savefile);
 
 
 // initialize vectors for computation
@@ -58,7 +58,7 @@ int ns_step_rk2( _Complex double* u, int K1, int K2, int N1, int N2, double nu,
 // Runge-Kutta-Fehlberg
 int ns_step_rkf45( _Complex double* u, double tolerance, double factor, 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);
 // Runge-Kutta-Dromand-Prince
-int ns_step_rkdp54( _Complex double* u, double tolerance, double factor, 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, 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);
 // Runge-Kutta-Bogacki-Shampine
 int ns_step_rkbs32( _Complex double* u, double tolerance, double factor, 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);