Fix MATSIZE

Makefile

@@ -30,7 +30,7 @@ LD=$(CC)
 #INCLUDES = 
 
 #LIBDIRS = 
-LIBS = -lm -lfftw3 -lfftw3_threads
+LIBS = -lm -lfftw3 -lfftw3_threads -lopenblas_64
 
 
 override LDFLAGS +=$(LIBDIRS)$(LIBS)

docs/nstrophy_doc.tex

@@ -15,6 +15,7 @@
 \documentclass{ian}
 
 \usepackage{largearray}
+\usepackage{dsfont}
 
 \begin{document}
 
@@ -86,6 +87,7 @@ and $q\cdot p^\perp$ is antisymmetric under exchange of $q$ and $p$. Therefore,
   \partial_t\hat u_k=
   -\frac{4\pi^2}{L^2}\nu k^2\hat u_k+\hat g_k
   +\frac{4\pi^2}{L^2|k|}T(\hat u,k)
+  =:\mathfrak F_k(\hat u)
   \label{ins_k}
 \end{equation}
 with
@@ -101,9 +103,8 @@ We truncate the Fourier modes and assume that $\hat u_k=0$ if $|k_1|>K_1$ or $|k
   \mathcal K:=\{(k_1,k_2),\ |k_1|\leqslant K_1,\ |k_2|\leqslant K_2\}
   .
 \end{equation}
-\bigskip
 
-\point{\bf Runge-Kutta methods}.
+\subsubsection{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:
@@ -198,9 +199,8 @@ It can be made by specifying the parameter {\tt adaptive\_norm}.
   \end{equation}
   This norm is selected by choosing {\tt adaptive\_norm=enstrophy}.
 \end{itemize}
-\bigskip
 
-\point{\bf Reality}.
+\subsubsection{Reality}.
 Since $U$ is real, $\hat U_{-k}=\hat U_k^*$, and so
 \begin{equation}
   \hat u_{-k}=\hat u_k^*
@@ -222,9 +222,8 @@ Thus,
   \label{realT}
 \end{equation}
 In order to keep the computation as quick as possible, we only compute and store the values for $k_1\geqslant 0$.
-\bigskip
 
-\point{\bf FFT}. We compute T using a fast Fourier transform, defined as
+\subsubsection{FFT}. We compute T using a fast Fourier transform, defined as
 \begin{equation}
   \mathcal F(f)(n):=\sum_{m\in\mathcal N}e^{-\frac{2i\pi}{N_1}m_1n_1-\frac{2i\pi}{N_2}m_2n_2}f(m_1,m_2)
 \end{equation}
@@ -267,9 +266,8 @@ Therefore,
     \mathcal F\left(q_x|q|\hat u_q\right)(n)
   \right)(k)
 \end{equation}
-\bigskip
 
-\point{\bf Energy}.
+\subsubsection{Energy}.
 We define the energy as
 \begin{equation}
   E(t)=\frac12\int\frac{dx}{L^2}\ U^2(t,x)=\frac12\sum_{k\in\mathbb Z^2}|\hat U_k|^2
@@ -370,18 +368,83 @@ and
   .
   \label{enstrophy}
 \end{equation}
-\bigskip
 
-\point{\bf Enstrophy}.
+\subsubsection{Enstrophy}.
 The enstrophy is defined as
 \begin{equation}
   \mathcal En(t)=\int\frac{dx}{L^2}\ |\nabla U|^2
   =\frac{4\pi^2}{L^2}\sum_{k\in\mathbb Z^2}k^2|\hat U_k|^2
   .
 \end{equation}
+
+\subsubsection{Lyapunov exponents}
+\indent
+To compute the Lyapunov exponents, we must first compute the Jacobian of $\hat u^{(n)}\mapsto\hat u^{(n+1)}$.
+This map is always of Runge-Kutta type, that is,
+\begin{equation}
+  \hat u^{(n+1)}=\hat u^{(n)}+\delta\sum_{i=1}^q w_i\mathfrak F(\hat u^{(n)})
+\end{equation}
+(see\-~(\ref{ins_k})), so
+\begin{equation}
+  D\hat u^{(n+1)}=\mathds 1+\delta\sum_{i=1}^q w_iD\mathfrak F(\hat u^{(n)})
+  .
+\end{equation}
+We then compute
+\begin{equation}
+  (D\mathfrak F(\hat u))_{k,\ell}
+  =
+  -\frac{4\pi^2}{L^2}\nu k^2\delta_{k,\ell}
+  +\frac{4\pi^2}{L^2|k|}\partial_{\hat u_\ell}T(\hat u,k)
+\end{equation}
+and, by\-~(\ref{T}),
+\begin{equation}
+  \partial_{\hat u_\ell}T(\hat u,k)
+  =
+  \sum_{\displaystyle\mathop{\scriptstyle q\in\mathbb Z^2}_{\ell+q=k}}
+  \left(
+    \frac{(q\cdot \ell^\perp)|q|}{|\ell|}
+    +
+    \frac{(\ell\cdot q^\perp)|\ell|}{|q|}
+  \right)\hat u_q
+  =
+  (k\cdot \ell^\perp)\left(
+    \frac{|k-\ell|}{|\ell|}
+    -
+    \frac{|\ell|}{|k-\ell|}
+  \right)\hat u_{k-\ell}
+  .
+\end{equation}
 \bigskip
 
-\point{\bf Numerical instability}.
+\indent
+The Lyapunov exponents are then defined as
+\begin{equation}
+  \frac1n\log\mathrm{spec}(\pi_i)
+  ,\quad
+  \pi_i:=\prod_{i=1}^nD\hat u^{(i)}
+  .
+\end{equation}
+However, the product of $D\hat u$ may become quite large or quite small (if the exponents are not all 1).
+To avoid this, we periodically rescale the product.
+We set $\mathfrak L_r\in\mathbb N_*$ (set by adjusting the {\tt lyanpunov\_reset} parameter), and, when $n$ is a multiple of $\mathfrak L_r$, we rescale the eigenvalues of $\pi_i$ to 1.
+To do so, we perform a $QR$ decomposition:
+\begin{equation}
+  \pi_{\alpha\mathfrak L_r}
+  =Q^{(\alpha)}R^{(\alpha)}
+\end{equation}
+where $Q^{(\alpha)}$ is diagonal and $R^{(\alpha)}$ is an orthogonal matrix, and we divide by $Q^{(\alpha)}$ (thus only keeping $R^{(\alpha)}$).
+Thus, we replace
+\begin{equation}
+  \pi_{\alpha\mathfrak L_r+i}\mapsto R^{(\alpha)}\prod_{j=1}^iD\hat u^{(j)}
+  .
+\end{equation}
+The Lyapunov exponents at time $\alpha\mathfrak L_r$ are then
+\begin{equation}
+  \frac1{\alpha\mathfrak L_r}\sum_{\beta=1}^\alpha\log\mathrm{spec}(Q^{(\beta)})
+  .
+\end{equation}
+
+\subsubsection{Numerical instability}.
 In order to prevent the algorithm from blowing up, it is necessary to impose the reality of $u(x)$ by hand, otherwise, truncation errors build up, and lead to divergences.
 It is sufficient to ensure that the convolution term $T(\hat u,k)$ satisfies $T(\hat u,-k)=T(\hat u,k)^*$.
 After imposing this condition, the algorithm no longer blows up, but it is still unstable (for instance, increasing $K_1$ or $K_2$ leads to very different results).

src/constants.cpp

@@ -14,11 +14,13 @@ See the License for the specific language governing permissions and
 limitations under the License.
 */
 
+#define M_PI 3.14159265358979323846
 
 #define COMMAND_UK 1
 #define COMMAND_ENSTROPHY 2
 #define COMMAND_QUIET 3
 #define COMMAND_RESUME 4
+#define COMMAND_LYAPUNOV 5
 
 #define DRIVING_ZERO 1
 #define DRIVING_TEST 2

src/lyapunov.c

@@ -0,0 +1,340 @@
+/*
+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 "constants.cpp"
+#include "lyapunov.h"
+#include <openblas64/cblas.h>
+#include <openblas64/lapacke.h>
+#include <math.h>
+#include <stdlib.h>
+
+#define MATSIZE (K1*(2*K2+1)+K2)
+
+// compute Lyapunov exponents
+int lyapunov(
+  int K1,
+  int K2,
+  int N1,
+  int N2,
+  double final_time,
+  double lyapunov_reset,
+  double nu,
+  double D_epsilon,
+  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 starting_time,
+  unsigned int nthreads
+){
+  double* lyapunov;
+  double* lyapunov_avg;
+  _Complex double* Du_prod;
+  _Complex double* Du;
+  _Complex double* u;
+  _Complex double* prevu;
+  _Complex double* tmp1;
+  _Complex double* tmp2;
+  _Complex double* tmp3;
+  _Complex double* tmp4;
+  _Complex double* tmp5;
+  _Complex double* tmp6;
+  _Complex double* tmp7;
+  _Complex double* tmp8;
+  _Complex double* tmp9;
+  _Complex double* tmp10;
+  double time;
+  fft_vect fft1;
+  fft_vect fft2;
+  fft_vect ifft;
+
+  // period
+  // add 0.1 to ensure proper rounding
+  uint64_t n=(uint64_t)((starting_time-fmod(starting_time, lyapunov_reset))/lyapunov_reset+0.1);
+
+  lyapunov_init_tmps(&lyapunov, &lyapunov_avg, &Du_prod, &Du, &u, &prevu, &tmp1, &tmp2, &tmp3, &tmp4, &tmp5, &tmp6, &tmp7, &tmp8, &tmp9, &tmp10, &fft1, &fft2, &ifft, K1, K2, N1, N2, nthreads, algorithm);
+  // copy initial condition
+  copy_u(u, u0, K1, K2);
+
+  // store step (useful for adaptive step methods
+  double prev_step=delta;
+  double step=delta;
+  double next_step=step;
+
+  const _Complex double one=1;
+  const _Complex double zero=0;
+
+  int i;
+  // init average
+  for (i=0; i<MATSIZE; i++){
+    lyapunov_avg[i]=0;
+  }
+
+  // save times at which Lyapunov exponents are computed
+  double prevtime=starting_time;
+
+  // iterate
+  time=starting_time;
+  while(final_time<0 || time<final_time){
+    // copy before step
+    copy_u(prevu, u, K1, K2);
+    prev_step=step;
+
+    ns_step(algorithm, u, K1, K2, N1, N2, nu, &step, &next_step, adaptive_tolerance, adaptive_factor, max_delta, adaptive_norm, L, g, time, starting_time, fft1, fft2, ifft, &tmp1, &tmp2, tmp3, tmp4, tmp5, tmp6, tmp7, irreversible, keep_en_cst, target_en);
+
+    // compute Jacobian
+    // do not touch tmp1, it might be used in the next step
+    ns_D_step(Du, prevu, u, K1, K2, N1, N2, nu, D_epsilon, prev_step, algorithm, adaptive_tolerance, adaptive_factor, max_delta, adaptive_norm, L, g, time, fft1, fft2, ifft, tmp2, tmp3, tmp4, tmp5, tmp6, tmp7, tmp8, tmp9, irreversible, keep_en_cst, target_en);
+
+    // multiply Jacobian
+    // switch to column major order, so transpose Du
+    cblas_zgemm(CblasColMajor, CblasNoTrans, CblasTrans, MATSIZE, MATSIZE, MATSIZE, &one, Du_prod, MATSIZE, Du, MATSIZE, &zero, tmp10, MATSIZE);
+    // copy back to Du_prod
+    _Complex double* move=tmp10;
+    tmp10=Du_prod;
+    Du_prod=move;
+
+    // increment time
+    time+=step;
+
+    // reset Jacobian
+    if(time>(n+1)*lyapunov_reset){
+      n++;
+
+      // QR decomposition
+      // do not touch tmp1, it might be used in the next step
+      LAPACKE_zgerqf(LAPACK_COL_MAJOR, MATSIZE, MATSIZE, Du_prod, MATSIZE, tmp2);
+      // extract eigenvalues (diagonal elements of Du_prod)
+      for(i=0; i<MATSIZE; i++){
+	lyapunov[i]=log(cabs(Du_prod[i*MATSIZE+i]))/(time-prevtime);
+      }
+
+      // sort lyapunov exponents
+      qsort(lyapunov, MATSIZE, sizeof(double), compare_double);
+
+      // average lyapunov
+      for(i=0; i<MATSIZE; i++){
+	// exclude inf
+	if((! isinf(lyapunov[i])) && (time>starting_time)){
+	  lyapunov_avg[i]=lyapunov_avg[i]*(prevtime-starting_time)/(time-starting_time)+lyapunov[i]*(time-prevtime)/(time-starting_time);
+	}
+      }
+
+      // print
+      for(i=0; i<MATSIZE; i++){
+	printf("% .15e % .15e % .15e\n",time, lyapunov[i], lyapunov_avg[i]);
+      }
+      printf("\n");
+      fprintf(stderr,"% .15e",time);
+      // print largest and smallest lyapunov exponent
+      if(MATSIZE>0){
+	fprintf(stderr," % .15e % .15e\n", lyapunov[0], lyapunov[MATSIZE-1]);
+      }
+
+      // set Du_prod to Q:
+      LAPACKE_zungrq(LAPACK_COL_MAJOR, MATSIZE, MATSIZE, MATSIZE, Du_prod, MATSIZE, tmp2);
+
+      // reset prevtime
+      prevtime=time;
+    }
+  }
+
+  lyapunov_free_tmps(lyapunov, lyapunov_avg, Du_prod, Du, u, prevu, tmp1, tmp2, tmp3, tmp4, tmp5, tmp6, tmp7, tmp8, tmp9, tmp10, fft1, fft2, ifft, algorithm);
+  return(0);
+}
+
+// comparison function for qsort
+int compare_double(const void* x, const void* y) {
+  if (*(const double*)x<*(const double*)y) {
+    return(-1);
+  } else if (*(const double*)x>*(const double*)y) {
+    return(+1);
+  } else {
+    return(0);
+  }
+}
+
+// Jacobian of u_n -> u_{n+1}
+int ns_D_step(
+  _Complex double* out,
+  _Complex double* un,
+  _Complex double* un_next,
+  int K1,
+  int K2,
+  int N1,
+  int N2,
+  double nu,
+  double epsilon,
+  double delta,
+  unsigned int algorithm,
+  double adaptive_tolerance,
+  double adaptive_factor,
+  double max_delta,
+  unsigned int adaptive_norm,
+  double L,
+  _Complex double* g,
+  double time,
+  fft_vect fft1,
+  fft_vect fft2,
+  fft_vect ifft,
+  _Complex double* tmp1,
+  _Complex double* tmp2,
+  _Complex double* tmp3,
+  _Complex double* tmp4,
+  _Complex double* tmp5,
+  _Complex double* tmp6,
+  _Complex double* tmp7,
+  _Complex double* tmp8,
+  bool irreversible,
+  bool keep_en_cst,
+  double target_en
+){
+  int lx,ly;
+  int i;
+  double step, next_step;
+
+  for(lx=0;lx<=K1;lx++){
+    for(ly=(lx>0 ? -K2 : 1);ly<=K2;ly++){
+      // derivative in the real direction
+      // finite difference vector
+      for (i=0;i<MATSIZE;i++){
+	if(i==klookup_sym(lx,ly,K2)){
+	  tmp1[i]=un[i]+epsilon;
+	}else{
+	  tmp1[i]=un[i];
+	}
+      }
+      // compute step 
+      // reinitialize step
+      step=delta;
+      next_step=delta;
+      ns_step(algorithm, tmp1, K1, K2, N1, N2, nu, &step, &next_step, adaptive_tolerance, adaptive_factor, max_delta, adaptive_norm, L, g, time, time, fft1, fft2, ifft, &tmp2, &tmp3, tmp4, tmp5, tmp6, tmp7, tmp8, irreversible, keep_en_cst, target_en);
+      // compute derivative
+      for (i=0;i<MATSIZE;i++){
+	// use row major order
+	out[klookup_sym(lx,ly,K2)*MATSIZE+i]=(tmp1[i]-un_next[i])/epsilon;
+      }
+
+      // derivative in the imaginary direction
+      // finite difference vector
+      for (i=0;i<MATSIZE;i++){
+	if(i==klookup_sym(lx,ly,K2)){
+	  tmp1[i]=un[i]+epsilon*I;
+	}else{
+	  tmp1[i]=un[i];
+	}
+      }
+      // compute step 
+      // reinitialize step
+      step=delta;
+      next_step=delta;
+      ns_step(algorithm, tmp1, K1, K2, N1, N2, nu, &step, &next_step, adaptive_tolerance, adaptive_factor, max_delta, adaptive_norm, L, g, time, time, fft1, fft2, ifft, &tmp2, &tmp3, tmp4, tmp5, tmp6, tmp7, tmp8, irreversible, keep_en_cst, target_en);
+      // compute derivative
+      for (i=0;i<MATSIZE;i++){
+	// use row major order
+	out[klookup_sym(lx,ly,K2)*MATSIZE+i]=-(tmp1[i]-un_next[i])/epsilon*I;
+      }
+    }
+  }
+
+  return(0);
+}
+
+
+int lyapunov_init_tmps(
+  double ** lyapunov,
+  double ** lyapunov_avg,
+  _Complex double ** Du_prod,
+  _Complex double ** Du,
+  _Complex double ** u,
+  _Complex double ** prevu,
+  _Complex double ** tmp1,
+  _Complex double ** tmp2,
+  _Complex double ** tmp3,
+  _Complex double ** tmp4,
+  _Complex double ** tmp5,
+  _Complex double ** tmp6,
+  _Complex double ** tmp7,
+  _Complex double ** tmp8,
+  _Complex double ** tmp9,
+  _Complex double ** tmp10,
+  fft_vect* fft1,
+  fft_vect* fft2,
+  fft_vect* ifft,
+  int K1,
+  int K2,
+  int N1,
+  int N2,
+  unsigned int nthreads,
+  unsigned int algorithm
+){
+  ns_init_tmps(u, tmp3, tmp4, tmp5, tmp6, tmp7, tmp8, tmp9, fft1, fft2, ifft, K1, K2, N1, N2, nthreads, algorithm);
+
+  *lyapunov=calloc(sizeof(double),MATSIZE);
+  *lyapunov_avg=calloc(sizeof(double),MATSIZE);
+  *Du_prod=calloc(sizeof(_Complex double),MATSIZE*MATSIZE);
+  *Du=calloc(sizeof(_Complex double),MATSIZE*MATSIZE);
+  *prevu=calloc(sizeof(_Complex double),MATSIZE);
+  *tmp1=calloc(sizeof(_Complex double),MATSIZE);
+  *tmp2=calloc(sizeof(_Complex double),MATSIZE);
+  *tmp10=calloc(sizeof(_Complex double),MATSIZE*MATSIZE);
+
+  return(0);
+}
+
+// release vectors
+int lyapunov_free_tmps(
+  double* lyapunov,
+  double* lyapunov_avg,
+  _Complex double* Du_prod,
+  _Complex double* Du,
+  _Complex double* u,
+  _Complex double* prevu,
+  _Complex double* tmp1,
+  _Complex double* tmp2,
+  _Complex double* tmp3,
+  _Complex double* tmp4,
+  _Complex double* tmp5,
+  _Complex double* tmp6,
+  _Complex double* tmp7,
+  _Complex double* tmp8,
+  _Complex double* tmp9,
+  _Complex double* tmp10,
+  fft_vect fft1,
+  fft_vect fft2,
+  fft_vect ifft,
+  unsigned int algorithm
+){
+  free(tmp10);
+  free(tmp2);
+  free(tmp1);
+  free(prevu);
+  free(Du);
+  free(Du_prod);
+  free(lyapunov_avg);
+  free(lyapunov);
+  ns_free_tmps(u, tmp3, tmp4, tmp5, tmp6, tmp7, tmp8, tmp9, fft1, fft2, ifft, algorithm);
+
+  return(0);
+}

src/lyapunov.h

@@ -0,0 +1,38 @@
+/*
+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 LYAPUNOV_H
+#define LYAPUNOV_H
+
+#include "navier-stokes.h"
+
+// compute Lyapunov exponents
+int lyapunov( int K1, int K2, int N1, int N2, double final_time, double lyapunov_reset, double nu, double D_epsilon, 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 starting_time, unsigned int nthreads);
+
+// comparison function for qsort
+int compare_double(const void* x, const void* y);
+
+// Jacobian of step
+int ns_D_step( _Complex double* out, _Complex double* un, _Complex double* un_next, int K1, int K2, int N1, int N2, double nu, double epsilon, double delta, unsigned int algorithm, double adaptive_tolerance, double adaptive_factor, double max_delta, unsigned int adaptive_norm, double L, _Complex double* g, double time, fft_vect fft1, fft_vect fft2, fft_vect ifft, _Complex double* tmp1, _Complex double* tmp2, _Complex double* tmp3, _Complex double* tmp4, _Complex double* tmp5, _Complex double* tmp6, _Complex double* tmp7, _Complex double* tmp8, bool irreversible, bool keep_en_cst, double target_en);
+
+// init vectors
+int lyapunov_init_tmps(double** lyapunov, double** lyapunov_avg, _Complex double ** Du_prod, _Complex double ** Du, _Complex double ** u, _Complex double ** prevu, _Complex double ** tmp1, _Complex double ** tmp2, _Complex double ** tmp3, _Complex double ** tmp4, _Complex double ** tmp5, _Complex double ** tmp6, _Complex double ** tmp7, _Complex double ** tmp8, _Complex double ** tmp9, _Complex double** tmp10, fft_vect* fft1, fft_vect* fft2, fft_vect* ifft, int K1, int K2, int N1, int N2, unsigned int nthreads, unsigned int algorithm);
+// release vectors
+int lyapunov_free_tmps(double* lyapunov, double* lyapunov_avg, _Complex double* Du_prod, _Complex double* Du, _Complex double* u, _Complex double* prevu, _Complex double* tmp1, _Complex double* tmp2, _Complex double* tmp3, _Complex double* tmp4, _Complex double* tmp5, _Complex double* tmp6, _Complex double* tmp7, _Complex double* tmp8, _Complex double* tmp9, _Complex double* tmp10, fft_vect fft1, fft_vect fft2, fft_vect ifft, unsigned int algorithm);
+
+#endif
+
+

src/main.c

@@ -27,6 +27,7 @@ limitations under the License.
 #include "dstring.h"
 #include "init.h"
 #include "int_tools.h"
+#include "lyapunov.h"
 #include "navier-stokes.h"
 
 
@@ -55,6 +56,8 @@ typedef struct nstrophy_parameters {
   bool keep_en_cst;
   FILE* initfile;
   FILE* drivingfile;
+  double lyapunov_reset;
+  double D_epsilon;
 } nstrophy_parameters;
 
 // usage message
@@ -279,6 +282,9 @@ int main (
   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.keep_en_cst, parameters.init_en, parameters.algorithm, nthreads, savefile);
   }
+  else if(command==COMMAND_LYAPUNOV){
+    lyapunov(parameters.K1, parameters.K2, parameters.N1, parameters.N2, parameters.final_time, parameters.lyapunov_reset, parameters.nu, parameters.D_epsilon, 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.starting_time, nthreads);
+  }
   else if(command==0){
     fprintf(stderr, "error: no command specified\n");
     print_usage();
@@ -501,6 +507,9 @@ int read_args(
 	*command=COMMAND_RESUME;
 	flag=CP_FLAG_RESUME;
       }
+      else if(strcmp(argv[i], "lyapunov")==0){
+	*command=COMMAND_LYAPUNOV;
+      }
       else{
 	fprintf(stderr, "error: unrecognized command: '%s'\n",argv[i]);
 	return(-1);
@@ -786,6 +795,20 @@ int set_parameter(
       return(-1);
     }
   }
+  else if (strcmp(lhs,"lyapunov_reset")==0){
+    ret=sscanf(rhs,"%lf",&(parameters->lyapunov_reset));
+    if(ret!=1){
+      fprintf(stderr, "error: parameter 'lyapunov_reset' should be a double\n       got '%s'\n",rhs);
+      return(-1);
+    }
+  }
+  else if (strcmp(lhs,"D_epsilon")==0){
+    ret=sscanf(rhs,"%lf",&(parameters->D_epsilon));
+    if(ret!=1){
+      fprintf(stderr, "error: parameter 'D_epsilon' should be a double\n       got '%s'\n",rhs);
+      return(-1);
+    }
+  }
   else if (strcmp(lhs,"driving")==0){
     if (strcmp(rhs,"zero")==0){
       parameters->driving=DRIVING_ZERO;

src/navier-stokes.c

@@ -89,21 +89,8 @@ int uk(
   // iterate
   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, 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, 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, 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, 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, keep_en_cst, target_en, time==starting_time);
-    } else {
-      fprintf(stderr,"bug: unknown algorithm: %u, contact ian.jauslin@rutgers,edu\n",algorithm);
-    }
+    // step
+    ns_step(algorithm, u, K1, K2, N1, N2, nu, &step, &next_step, adaptive_tolerance, adaptive_factor, max_delta, adaptive_norm, L, g, time, starting_time, fft1, fft2, ifft, &tmp1, &tmp2, tmp3, tmp4, tmp5, tmp6, tmp7, irreversible, keep_en_cst, target_en);
 
     time+=step;
     step=next_step;
@@ -205,21 +192,7 @@ int enstrophy(
   // iterate
   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, 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, 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, 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, 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, keep_en_cst, target_en, time==starting_time);
-    } else {
-      fprintf(stderr,"bug: unknown algorithm: %u, contact ian.jauslin@rutgers,edu\n",algorithm);
-    }
+    ns_step(algorithm, u, K1, K2, N1, N2, nu, &step, &next_step, adaptive_tolerance, adaptive_factor, max_delta, adaptive_norm, L, g, time, starting_time, fft1, fft2, ifft, &tmp1, &tmp2, tmp3, tmp4, tmp5, tmp6, tmp7, irreversible, keep_en_cst, target_en);
 
     time+=step;
 
@@ -386,21 +359,7 @@ int quiet(
   // iterate
   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, 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, 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, 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, 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, keep_en_cst, target_en, time==starting_time);
-    } else {
-      fprintf(stderr,"bug: unknown algorithm: %u, contact ian.jauslin@rutgers,edu\n",algorithm);
-    }
+    ns_step(algorithm, u, K1, K2, N1, N2, nu, &step, &next_step, adaptive_tolerance, adaptive_factor, max_delta, adaptive_norm, L, g, time, starting_time, fft1, fft2, ifft, &tmp1, &tmp2, tmp3, tmp4, tmp5, tmp6, tmp7, irreversible, keep_en_cst, target_en);
 
     time+=step;
     step=next_step;
@@ -553,6 +512,60 @@ int copy_u(
 }
 
 // next time step
+int ns_step(
+  unsigned int algorithm,
+  _Complex double* u,
+  int K1,
+  int K2,
+  int N1,
+  int N2,
+  double nu,
+  double* delta,
+  double* next_delta,
+  double adaptive_tolerance,
+  double adaptive_factor,
+  double max_delta,
+  unsigned int adaptive_norm,
+  double L,
+  _Complex double* g,
+  double time,
+  double starting_time,
+  fft_vect fft1,
+  fft_vect fft2,
+  fft_vect ifft,
+  // the pointers tmp1 and tmp2 will be exchanged at the end of the routine for first-same-as-last RK algorithms
+  _Complex double** tmp1,
+  _Complex double** tmp2,
+  _Complex double* tmp3,
+  _Complex double* tmp4,
+  _Complex double* tmp5,
+  _Complex double* tmp6,
+  _Complex double* tmp7,
+  bool irreversible,
+  bool keep_en_cst,
+  double target_en
+){
+  if(algorithm==ALGORITHM_RK2){
+    ns_step_rk2(u, K1, K2, N1, N2, nu, *delta, 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, *delta, 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, delta, next_delta, 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
+    // first-same-as-last with 2-nd argument
+    ns_step_rkdp54(u, adaptive_tolerance, adaptive_factor, max_delta, adaptive_norm, K1, K2, N1, N2, nu, delta, next_delta, 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
+    // first-same-as-last with 4-th argument
+    ns_step_rkbs32(u, adaptive_tolerance, adaptive_factor, max_delta, adaptive_norm, K1, K2, N1, N2, nu, delta, next_delta, L, g, fft1, fft2, ifft, tmp1, tmp3, tmp4, tmp2, tmp5, irreversible, keep_en_cst, target_en, time==starting_time);
+  } else {
+    fprintf(stderr,"bug: unknown algorithm: %u, contact ian.jauslin@rutgers.edu\n",algorithm);
+  }
+
+  return (0);
+}
+
 // RK 4 algorithm
 int ns_step_rk4(
   _Complex double* u,
@@ -1342,6 +1355,80 @@ int ns_T_nofft(
   return 0;
 }
 
+
+/*
+// Jacobian of rhs
+int ns_D_rhs(
+  _Complex double* out,
+  _Complex double* u,
+  int K1,
+  int K2,
+  double nu,
+  double L,
+  bool irreversible
+){
+  int kx,ky,lx,ly;
+  int i;
+  double alpha;
+
+  if (irreversible) {
+    alpha=nu;
+  } else {
+    // TODO
+    alpha=0;
+  }
+
+  for(i=0; i<MATSIZE*MATSIZE; i++){
+    out[i]=0;
+  }
+  // diagonal term
+  for(kx=0;kx<=K1;kx++){
+    for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
+      out[klookup_sym(kx,ky,K2)*MATSIZE+klookup_sym(kx,ky,K2)]=-4*M_PI*M_PI/L/L*alpha*(kx*kx+ky*ky);
+    }
+  }
+  // convolution term
+  for(kx=0;kx<=K1;kx++){
+    for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
+      for(lx=0;lx<=K1;lx++){
+	for(ly=(lx>0 ? -K2 : 1);ly<=K2;ly++){
+	  // no contribution from k=l
+	  if (kx!=lx && ky!=ly){
+	    // use row-major ordering
+	    out[klookup_sym(kx,ky,K2)*MATSIZE+klookup_sym(lx,ly,K2)]=4*M_PI*M_PI/L/L*ns_d_T(u,kx,ky,lx,ly,K2);
+	  }
+	}
+      }
+    }
+  }
+
+  return(0);
+}
+
+// derivative of T with respect to u_k
+_Complex double ns_d_T(
+  _Complex double* u,
+  int kx,
+  int ky,
+  int lx,
+  int ly,
+  int K2
+){
+  int qx=kx-lx;
+  int qy=ky-ly;
+  // trivial case
+  if (qx*qx+qy*qy==0){
+    return 0.;
+  }
+  if (qx>0 || (qx==0 && qy>=1)){
+    return (ky*lx-kx*ly)*(qx*qx+qy*qy-lx*lx-ly*ly)/sqrt(lx*lx+ly*ly)/sqrt(qx*qx+qy*qy)*u[klookup_sym(qx,qy,K2)];
+  }else{
+    return (ky*lx-kx*ly)*(qx*qx+qy*qy-lx*lx-ly*ly)/sqrt(lx*lx+ly*ly)/sqrt(qx*qx+qy*qy)*u[klookup_sym(-qx,-qy,K2)];
+  }
+}
+*/
+
+
 // compute alpha
 double compute_alpha(
   _Complex double* u,

src/navier-stokes.h

@@ -21,8 +21,6 @@ limitations under the License.
 #include <stdbool.h>
 #include <fftw3.h>
 
-#define M_PI 3.14159265358979323846
-
 // abort signal
 extern volatile bool g_abort;
 
@@ -51,6 +49,7 @@ int ns_free_tmps( _Complex double* u, _Complex double* tmp1, _Complex double *tm
 int copy_u( _Complex double* u, _Complex double* u0, int K1, int K2);
 
 // next time step for Irreversible/reversible Navier-Stokes equation
+int ns_step( unsigned int algorithm, _Complex double* u, int K1, int K2, int N1, int N2, double nu, double* delta, double* next_delta, double adaptive_tolerance, double adaptive_factor, double max_delta, unsigned int adaptive_norm, double L, _Complex double* g, double time, double starting_time, fft_vect fft1, fft_vect fft2, fft_vect ifft, _Complex double** tmp1, _Complex double** tmp2, _Complex double* tmp3, _Complex double* tmp4, _Complex double* tmp5, _Complex double* tmp6, _Complex double* tmp7, bool irreversible, bool keep_en_cst, double target_en);
 // 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, bool keep_en_cst, double target_en);
 // RK2
@@ -71,6 +70,13 @@ int ns_T( _Complex double* u, int K1, int K2, int N1, int N2, fft_vect fft1, fft
 // convolution term in right side of equation (computed without fft)
 int ns_T_nofft( _Complex double* out, _Complex double* u, int K1, int K2, int N1, int N2);
 
+/*
+// Jacobian of rhs
+int ns_D_rhs( _Complex double* out, _Complex double* u, int K1, int K2, double nu, double L, bool irreversible);
+// derivative of T with respect to u_k
+_Complex double ns_d_T( _Complex double* u, int kx, int ky, int lx, int ly, int K2);
+*/
+
 // compute alpha
 double compute_alpha( _Complex double* u, int K1, int K2, _Complex double* g, double L);
 // compute energy