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Rewrite cost function for adaptive step

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This commit is contained in:
Ian Jauslin 2024-11-18 17:09:17 -05:00
parent 9fa10c8db4
commit d81ad18618
6 changed files with 211 additions and 264 deletions
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8
README.md
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@ -122,10 +122,10 @@ should be a `;` sperated list of `key=value` pairs. The possible keys are
* `max_delta` (double, default 1e-2): when using the adaptive step, do not * `max_delta` (double, default 1e-2): when using the adaptive step, do not
exceet `delta_max`. exceet `delta_max`.
* `adaptive_norm`: norm to use to estimate the error of the adaptive method: * `adaptive_cost`: cost function to use to estimate the error of the adaptive
`L1` (default) for the normalized L1 norm, `k3` for the normalized L1 norm of method: `L1` (default) for the normalized L1 norm, `k3` for the normalized L1
f_k/|k|^3, `k32` for the normalized L1 norm, `enstrophy` for the enstrophy norm of f_k/|k|^3, `k32` for the normalized L1 norm, `enstrophy` for the
norm. enstrophy, `alpha` for alpha.
* `keep_en_cst` (0 or 1, default 0): impose that the enstrophy is constant at * `keep_en_cst` (0 or 1, default 0): impose that the enstrophy is constant at
each step (only really useful for the reversible equation). each step (only really useful for the reversible equation).

85
docs/nstrophy_doc.tex
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@ -115,18 +115,18 @@ In addition, several variable step methods are implemented:
\end{itemize} \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: 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} \begin{equation}
\|\hat u^{(n)}-\hat U^{(n)}\| D(\hat u^{(n)},\hat U^{(n)})
<\epsilon_{\mathrm{target}} <\epsilon_{\mathrm{target}}
\end{equation} \end{equation}
for some given $\epsilon_{\mathrm{target}}$, set using the {\tt adaptive\_tolerance} parameter. for some given {\it cost function} $D$, and $\epsilon_{\mathrm{target}}$, set using the {\tt adaptive\_tolerance} parameter.
The choice of the norm matters, and will be discussed below. The choice of the cost function matters, and will be discussed below.
If the error is larger than the target, then the step size is decreased. If the error is larger than the target, then the step size is decreased.
How this is done depends on the order of algorithm. 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 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 (as long as the cost function is such that $D(\hat u,\hat u+\varphi)\sim\|\varphi\|$ in some norm)
\begin{equation} \begin{equation}
\|\hat u^{(n)}-\hat U^{(n)}\|=\delta_n^qC_n D(\hat u^{(n)},\hat U^{(n)})=\delta_n^qC_n
.
\end{equation} \end{equation}
for some number $C_n$.
We wish to set $\delta_{n+1}$ so that We wish to set $\delta_{n+1}$ so that
\begin{equation} \begin{equation}
\delta_{n+1}^qC_n=\epsilon_{\mathrm{target}} \delta_{n+1}^qC_n=\epsilon_{\mathrm{target}}
@ -135,7 +135,7 @@ so
\begin{equation} \begin{equation}
\delta_{n+1} \delta_{n+1}
=\left(\frac{\epsilon_{\mathrm{target}}}{C_n}\right)^{\frac1q} =\left(\frac{\epsilon_{\mathrm{target}}}{C_n}\right)^{\frac1q}
=\delta_n\left(\frac{\epsilon_{\mathrm{target}}}{\|\hat u^{(n)}-\hat U^{(n)}\|}\right)^{\frac1q} =\delta_n\left(\frac{\epsilon_{\mathrm{target}}}{D(\hat u^{(n)},\hat U^{(n)})}\right)^{\frac1q}
. .
\label{adaptive_delta} \label{adaptive_delta}
\end{equation} \end{equation}
@ -145,59 +145,49 @@ To be safe, we also set a maximal value for $\delta$ via the {\tt max\_delta} pa
\bigskip \bigskip
\indent \indent
The choice of the norm $\|\cdot\|$ matters. The choice of the cost function $D$ matters.
It can be made by specifying the parameter {\tt adaptive\_norm}. It can be made by specifying the parameter {\tt adaptive\_cost}.
\begin{itemize} \begin{itemize}
\item A naive choice is to take $\|\cdot\|$ to be the normalized $L_1$ norm: \item
For computations where the main focus is the enstrophy\-~(\ref{enstrophy}), one may want to set the cost function to the relative difference of the enstrophies:
\begin{equation} \begin{equation}
\|f\|:= D(\hat u,\hat U):=\frac{|\mathcal En(\hat u)-\mathcal En(\hat U)|}{\mathcal En(\hat u)}
\frac1{\mathcal N}\sum_k|f_k|
,\quad
\mathcal N:=\sum_k|\hat u_k^{(n)}-\hat u_k^{(n-1)}|
. .
\end{equation} \end{equation}
This norm is selected by choosing {\tt adaptive\_norm=L1}. This cost function is selected by choosing {\tt adaptive\_cost=enstrophy}.
\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 \item
For computations where the main focus is the value of $\alpha$\-~(\ref{alpha}), one may want to set the cost function to the relative difference of $\alpha$:
\begin{equation} \begin{equation}
\|f\|:=\frac1{\mathcal N}\sum_k|f_k|k^{-3} D(\hat u,\hat U):=\frac{|\alpha(\hat u)-\alpha(\hat U)|}{|\alpha(\hat u)|}
.
\end{equation}
This cost function is selected by choosing {\tt adaptive\_cost=alpha}.
\item Alternatively, one my take $D$ to be the normalized $L_1$ norm:
\begin{equation}
D(\hat u,\hat U):=
\frac1{\mathcal N}\sum_k|\hat u_k-\hat U_k|
,\quad ,\quad
\mathcal N:=\sum_k|\hat u_k^{(n)}-\hat u_k^{(n-1)}|k^{-3} \mathcal N:=\sum_k|\hat u_k|
.
\end{equation}
This function is selected by choosing {\tt adaptive\_cost=L1}.
\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 cost function of the form
\begin{equation}
D(\hat u,\hat U):=\frac1{\mathcal N}\sum_k|\hat u_k-\hat U_k|k^{-3}
,\quad
\mathcal N:=\sum_k|\hat u_k|k^{-3}
\end{equation} \end{equation}
or or
\begin{equation} \begin{equation}
\|f\|:=\frac1{\mathcal N}\sum_k|f_k|k^{-\frac32} D(\hat u,\hat U):=\frac1{\mathcal N}\sum_k|\hat u_k-\hat U_k|k^{-\frac32}
,\quad ,\quad
\mathcal N:=\sum_k|\hat u_k^{(n)}-\hat u_k^{(n-1)}|k^{-\frac32} \mathcal N:=\sum_k|\hat u_k|k^{-\frac32}
\end{equation} \end{equation}
are sensible choices. are sensible choices.
These norms are selected by choosing {\tt adaptive\_norm=k3} and {\tt adaptive\_norm=k32} respectively. These cost functions are selected by choosing {\tt adaptive\_cost=k3} and {\tt adaptive\_cost=k32} respectively.
\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:=\frac{\sqrt{\sum_k k^2|\hat u_k^{(n)}|^2}+\sqrt{\sum_k k^2|\hat U_k^{(n)}|^2}}{\sum_k k^2|\hat u_k^{(n)}|^2}
.
\end{equation}
Doing so controls the error of the enstrophy through
\begin{equation}
\frac1{\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}
so
\begin{equation}
\frac1{\mathcal N^2}
|\mathcal En(\hat u)-\mathcal En(\hat U)|\leqslant
\|\hat u-\hat U\|\frac1{\mathcal N}\left(\sqrt{\sum_k k^2|\hat u_k|^2}+\sqrt{\sum_k k^2|\hat U_k|^2}\right)
\end{equation}
and thus
\begin{equation}
\frac{|\mathcal En(\hat u)-\mathcal En(\hat U)|}{\mathcal En(\hat u)}\leqslant
\|\hat u-\hat U\|
.
\end{equation}
This norm is selected by choosing {\tt adaptive\_norm=enstrophy}.
\end{itemize} \end{itemize}
\subsubsection{Reality}. \subsubsection{Reality}.
@ -511,6 +501,7 @@ and so
\alpha(\hat u) \alpha(\hat u)
=\frac{\frac{L^2}{4\pi^2}\sum_k k^2\hat u_k^*\hat g_k+\sum_k|k|\hat u_k^*T(\hat u,k)}{\sum_kk^4|\hat u_k|^2} =\frac{\frac{L^2}{4\pi^2}\sum_k k^2\hat u_k^*\hat g_k+\sum_k|k|\hat u_k^*T(\hat u,k)}{\sum_kk^4|\hat u_k|^2}
. .
\label{alpha}
\end{equation} \end{equation}
Note that, by\-~(\ref{realu})-(\ref{realT}), Note that, by\-~(\ref{realu})-(\ref{realT}),
\begin{equation} \begin{equation}

9
src/constants.cpp
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@ -40,7 +40,8 @@ limitations under the License.
#define ALGORITHM_RKDP54 1002 #define ALGORITHM_RKDP54 1002
#define ALGORITHM_RKBS32 1003 #define ALGORITHM_RKBS32 1003
#define NORM_L1 1 #define COST_L1 1
#define NORM_k3 2 #define COST_k3 2
#define NORM_k32 3 #define COST_k32 3
#define NORM_ENSTROPHY 4 #define COST_ENSTROPHY 4
#define COST_ALPHA 5

48
src/main.c
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@ -46,7 +46,7 @@ typedef struct nstrophy_parameters {
double adaptive_tolerance; double adaptive_tolerance;
double adaptive_factor; double adaptive_factor;
double max_delta; double max_delta;
unsigned int adaptive_norm; unsigned int adaptive_cost;
double print_freq; double print_freq;
int seed; int seed;
double starting_time; double starting_time;
@ -272,19 +272,19 @@ int main (
// run command // run command
if (command==COMMAND_UK){ 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.keep_en_cst, parameters.init_en, 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_cost, 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){ else if(command==COMMAND_ENSTROPHY){
// register signal handler to handle aborts // register signal handler to handle aborts
signal(SIGINT, sig_handler); signal(SIGINT, sig_handler);
signal(SIGTERM, 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.keep_en_cst, parameters.init_en, parameters.algorithm, parameters.print_freq, parameters.starting_time, parameters.print_alpha, nthreads, savefile, utfile, (char*)argv[0], dstring_to_str_noinit(&param_str), dstring_to_str_noinit(&savefile_str), dstring_to_str_noinit(&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_cost, u0, g, parameters.irreversible, parameters.keep_en_cst, parameters.init_en, parameters.algorithm, parameters.print_freq, parameters.starting_time, parameters.print_alpha, nthreads, savefile, utfile, (char*)argv[0], dstring_to_str_noinit(&param_str), dstring_to_str_noinit(&savefile_str), dstring_to_str_noinit(&utfile_str));
} }
else if(command==COMMAND_QUIET){ 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); 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_cost, parameters.starting_time, u0, g, parameters.irreversible, parameters.keep_en_cst, parameters.init_en, parameters.algorithm, nthreads, savefile);
} }
else if(command==COMMAND_LYAPUNOV){ 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); 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_cost, u0, g, parameters.irreversible, parameters.keep_en_cst, parameters.init_en, parameters.algorithm, parameters.starting_time, nthreads);
} }
else if(command==0){ else if(command==0){
fprintf(stderr, "error: no command specified\n"); fprintf(stderr, "error: no command specified\n");
@ -393,18 +393,21 @@ int print_params(
} }
if(parameters.algorithm>ALGORITHM_ADAPTIVE_THRESHOLD){ if(parameters.algorithm>ALGORITHM_ADAPTIVE_THRESHOLD){
switch(parameters.adaptive_norm){ switch(parameters.adaptive_cost){
case NORM_L1: case COST_L1:
fprintf(file,", norm=L1"); fprintf(file,", cost=L1");
break; break;
case NORM_k3: case COST_k3:
fprintf(file,", norm=k3"); fprintf(file,", cost=k3");
break; break;
case NORM_k32: case COST_k32:
fprintf(file,", norm=k32"); fprintf(file,", cost=k32");
break; break;
case NORM_ENSTROPHY: case COST_ENSTROPHY:
fprintf(file,", norm=enstrophy"); fprintf(file,", cost=enstrophy");
break;
case COST_ALPHA:
fprintf(file,", cost=alpha");
break; break;
} }
} }
@ -544,7 +547,7 @@ int set_default_params(
parameters->adaptive_tolerance=1e-11; parameters->adaptive_tolerance=1e-11;
parameters->adaptive_factor=0.9; parameters->adaptive_factor=0.9;
parameters->max_delta=1e-2; parameters->max_delta=1e-2;
parameters->adaptive_norm=NORM_L1; parameters->adaptive_cost=COST_L1;
parameters->final_time=100000; parameters->final_time=100000;
parameters->print_freq=1; parameters->print_freq=1;
parameters->starting_time=0; parameters->starting_time=0;
@ -759,21 +762,24 @@ int set_parameter(
return(-1); return(-1);
} }
} }
else if (strcmp(lhs,"adaptive_norm")==0){ else if (strcmp(lhs,"adaptive_cost")==0){
if (strcmp(rhs,"L1")==0){ if (strcmp(rhs,"L1")==0){
parameters->adaptive_norm=NORM_L1; parameters->adaptive_cost=COST_L1;
} }
else if (strcmp(rhs,"k3")==0){ else if (strcmp(rhs,"k3")==0){
parameters->adaptive_norm=NORM_k3; parameters->adaptive_cost=COST_k3;
} }
else if (strcmp(rhs,"k32")==0){ else if (strcmp(rhs,"k32")==0){
parameters->adaptive_norm=NORM_k32; parameters->adaptive_cost=COST_k32;
} }
else if (strcmp(rhs,"enstrophy")==0){ else if (strcmp(rhs,"enstrophy")==0){
parameters->adaptive_norm=NORM_ENSTROPHY; parameters->adaptive_cost=COST_ENSTROPHY;
}
else if (strcmp(rhs,"alpha")==0){
parameters->adaptive_cost=COST_ALPHA;
} }
else{ else{
fprintf(stderr, "error: unrecognized adaptive_norm '%s'\n",rhs); fprintf(stderr, "error: unrecognized adaptive_cost '%s'\n",rhs);
return(-1); return(-1);
} }
} }

297
src/navier-stokes.c
View File

@ -36,7 +36,7 @@ int uk(
double adaptive_tolerance, double adaptive_tolerance,
double adaptive_factor, double adaptive_factor,
double max_delta, double max_delta,
unsigned int adaptive_norm, unsigned int adaptive_cost,
_Complex double* u0, _Complex double* u0,
_Complex double* g, _Complex double* g,
bool irreversible, bool irreversible,
@ -90,7 +90,7 @@ int uk(
time=starting_time; time=starting_time;
while(final_time<0 || time<final_time){ while(final_time<0 || time<final_time){
// 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); ns_step(algorithm, u, K1, K2, N1, N2, nu, &step, &next_step, adaptive_tolerance, adaptive_factor, max_delta, adaptive_cost, L, g, time, starting_time, fft1, fft2, ifft, &tmp1, &tmp2, tmp3, tmp4, tmp5, tmp6, tmp7, irreversible, keep_en_cst, target_en);
time+=step; time+=step;
step=next_step; step=next_step;
@ -136,7 +136,7 @@ int enstrophy(
double adaptive_tolerance, double adaptive_tolerance,
double adaptive_factor, double adaptive_factor,
double max_delta, double max_delta,
unsigned int adaptive_norm, unsigned int adaptive_cost,
_Complex double* u0, _Complex double* u0,
_Complex double* g, _Complex double* g,
bool irreversible, bool irreversible,
@ -193,7 +193,7 @@ int enstrophy(
// iterate // iterate
time=starting_time; time=starting_time;
while(final_time<0 || time<final_time){ while(final_time<0 || time<final_time){
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); ns_step(algorithm, u, K1, K2, N1, N2, nu, &step, &next_step, adaptive_tolerance, adaptive_factor, max_delta, adaptive_cost, L, g, time, starting_time, fft1, fft2, ifft, &tmp1, &tmp2, tmp3, tmp4, tmp5, tmp6, tmp7, irreversible, keep_en_cst, target_en);
time+=step; time+=step;
@ -281,7 +281,7 @@ int quiet(
double adaptive_tolerance, double adaptive_tolerance,
double adaptive_factor, double adaptive_factor,
double max_delta, double max_delta,
unsigned int adaptive_norm, unsigned int adaptive_cost,
double starting_time, double starting_time,
_Complex double* u0, _Complex double* u0,
_Complex double* g, _Complex double* g,
@ -317,7 +317,7 @@ int quiet(
// iterate // iterate
time=starting_time; time=starting_time;
while(final_time<0 || time<final_time){ while(final_time<0 || time<final_time){
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); ns_step(algorithm, u, K1, K2, N1, N2, nu, &step, &next_step, adaptive_tolerance, adaptive_factor, max_delta, adaptive_cost, L, g, time, starting_time, fft1, fft2, ifft, &tmp1, &tmp2, tmp3, tmp4, tmp5, tmp6, tmp7, irreversible, keep_en_cst, target_en);
time+=step; time+=step;
step=next_step; step=next_step;
@ -483,7 +483,7 @@ int ns_step(
double adaptive_tolerance, double adaptive_tolerance,
double adaptive_factor, double adaptive_factor,
double max_delta, double max_delta,
unsigned int adaptive_norm, unsigned int adaptive_cost,
double L, double L,
_Complex double* g, _Complex double* g,
double time, double time,
@ -508,15 +508,15 @@ int ns_step(
} else if (algorithm==ALGORITHM_RK4) { } 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); 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) { } 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); ns_step_rkf45(u, adaptive_tolerance, adaptive_factor, max_delta, adaptive_cost, 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) { } else if (algorithm==ALGORITHM_RKDP54) {
// only compute k1 on the first step // only compute k1 on the first step
// first-same-as-last with 2-nd argument // 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); ns_step_rkdp54(u, adaptive_tolerance, adaptive_factor, max_delta, adaptive_cost, 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) { } else if (algorithm==ALGORITHM_RKBS32) {
// only compute k1 on the first step // only compute k1 on the first step
// first-same-as-last with 4-th argument // 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); ns_step_rkbs32(u, adaptive_tolerance, adaptive_factor, max_delta, adaptive_cost, 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 { } else {
fprintf(stderr,"bug: unknown algorithm: %u, contact ian.jauslin@rutgers.edu\n",algorithm); fprintf(stderr,"bug: unknown algorithm: %u, contact ian.jauslin@rutgers.edu\n",algorithm);
} }
@ -674,7 +674,7 @@ int ns_step_rkf45(
double tolerance, double tolerance,
double factor, double factor,
double max_delta, double max_delta,
unsigned int adaptive_norm, unsigned int adaptive_cost,
int K1, int K1,
int K2, int K2,
int N1, int N1,
@ -701,7 +701,7 @@ int ns_step_rkf45(
bool compute_k1 bool compute_k1
){ ){
int kx,ky; int kx,ky;
double err,relative; double cost;
// k1: u(t) // k1: u(t)
if(compute_k1){ if(compute_k1){
@ -748,72 +748,29 @@ int ns_step_rkf45(
} }
ns_rhs(k6, tmp, K1, K2, N1, N2, nu, L, g, fft1, fft2, ifft, irreversible); ns_rhs(k6, tmp, K1, K2, N1, N2, nu, L, g, fft1, fft2, ifft, irreversible);
// compute error // next step
err=0; for(kx=0;kx<=K1;kx++){
if(adaptive_norm==NORM_L1){ for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
relative=0; // u
for(kx=0;kx<=K1;kx++){ tmp[klookup_sym(kx,ky,K2)]=u[klookup_sym(kx,ky,K2)]+(*delta)*(25./216*k1[klookup_sym(kx,ky,K2)]+1408./2565*k3[klookup_sym(kx,ky,K2)]+2197./4104*k4[klookup_sym(kx,ky,K2)]-1./5*k5[klookup_sym(kx,ky,K2)]);
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){ // U: save to k6, which is no longer needed
err+=cabs((*delta)*(1./360*k1[klookup_sym(kx,ky,K2)]-128./4275*k3[klookup_sym(kx,ky,K2)]-2197./75240*k4[klookup_sym(kx,ky,K2)]+1./50*k5[klookup_sym(kx,ky,K2)]+2./55*k6[klookup_sym(kx,ky,K2)])); k6[klookup_sym(kx,ky,K2)]=u[klookup_sym(kx,ky,K2)]+(*delta)*(16./135*k1[klookup_sym(kx,ky,K2)]+6656./12825*k3[klookup_sym(kx,ky,K2)]+28561./56430*k4[klookup_sym(kx,ky,K2)]-9./50*k5[klookup_sym(kx,ky,K2)]+2./55*k6[klookup_sym(kx,ky,K2)]);
// next step
tmp[klookup_sym(kx,ky,K2)]=(*delta)*(25./216*k1[klookup_sym(kx,ky,K2)]+1408./2565*k3[klookup_sym(kx,ky,K2)]+2197./4104*k4[klookup_sym(kx,ky,K2)]-1./5*k5[klookup_sym(kx,ky,K2)]);
relative+=cabs(tmp[klookup_sym(kx,ky,K2)]);
}
} }
} }
else if(adaptive_norm==NORM_k3){
relative=0; // cost function
for(kx=0;kx<=K1;kx++){ cost=ns_adaptive_cost(tmp,k6,adaptive_cost,K1,K2,g,L);
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
err+=cabs((*delta)*(1./360*k1[klookup_sym(kx,ky,K2)]-128./4275*k3[klookup_sym(kx,ky,K2)]-2197./75240*k4[klookup_sym(kx,ky,K2)]+1./50*k5[klookup_sym(kx,ky,K2)]+2./55*k6[klookup_sym(kx,ky,K2)]))/pow(kx*kx+ky*ky,1.5);
// next step
tmp[klookup_sym(kx,ky,K2)]=(*delta)*(25./216*k1[klookup_sym(kx,ky,K2)]+1408./2565*k3[klookup_sym(kx,ky,K2)]+2197./4104*k4[klookup_sym(kx,ky,K2)]-1./5*k5[klookup_sym(kx,ky,K2)]);
relative+=cabs(tmp[klookup_sym(kx,ky,K2)])/pow(kx*kx+ky*ky,1.5);
}
}
}
else if(adaptive_norm==NORM_k32){
relative=0;
for(kx=0;kx<=K1;kx++){
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
err+=cabs((*delta)*(1./360*k1[klookup_sym(kx,ky,K2)]-128./4275*k3[klookup_sym(kx,ky,K2)]-2197./75240*k4[klookup_sym(kx,ky,K2)]+1./50*k5[klookup_sym(kx,ky,K2)]+2./55*k6[klookup_sym(kx,ky,K2)]))/pow(kx*kx+ky*ky,0.75);
// next step
tmp[klookup_sym(kx,ky,K2)]=(*delta)*(25./216*k1[klookup_sym(kx,ky,K2)]+1408./2565*k3[klookup_sym(kx,ky,K2)]+2197./4104*k4[klookup_sym(kx,ky,K2)]-1./5*k5[klookup_sym(kx,ky,K2)]);
relative+=cabs(tmp[klookup_sym(kx,ky,K2)])/pow(kx*kx+ky*ky,0.75);
}
}
}
else if(adaptive_norm==NORM_ENSTROPHY){
double sumu, sumU;
sumu=0;
sumU=0;
for(kx=0;kx<=K1;kx++){
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
err+=(kx*kx+ky*ky)*cabs2((*delta)*(1./360*k1[klookup_sym(kx,ky,K2)]-128./4275*k3[klookup_sym(kx,ky,K2)]-2197./75240*k4[klookup_sym(kx,ky,K2)]+1./50*k5[klookup_sym(kx,ky,K2)]+2./55*k6[klookup_sym(kx,ky,K2)]));
// next step
tmp[klookup_sym(kx,ky,K2)]=(*delta)*(25./216*k1[klookup_sym(kx,ky,K2)]+1408./2565*k3[klookup_sym(kx,ky,K2)]+2197./4104*k4[klookup_sym(kx,ky,K2)]-1./5*k5[klookup_sym(kx,ky,K2)]);
sumU+=(kx*kx+ky*ky)*cabs2(u[klookup_sym(kx,ky,K2)]+(*delta)*(16./135*k1[klookup_sym(kx,ky,K2)]+6656./12825*k3[klookup_sym(kx,ky,K2)]+28561./56430*k4[klookup_sym(kx,ky,K2)]-9./50*k5[klookup_sym(kx,ky,K2)]+2./55*k6[klookup_sym(kx,ky,K2)]));
sumu+=(kx*kx+ky*ky)*cabs2(u[klookup_sym(kx,ky,K2)]+tmp[klookup_sym(kx,ky,K2)]);
}
}
err=sqrt(err);
relative=(sqrt(sumu)+sqrt(sumU))/sumu;
}
else{
fprintf(stderr,"bug: unknown norm: %u, contact ian.jauslin@rutgers.edu\n", adaptive_norm);
exit(-1);
}
// compare relative error with tolerance // compare relative error with tolerance
if(err<relative*tolerance){ if(cost<tolerance){
// add to output // copy to output
for(kx=0;kx<=K1;kx++){ for(kx=0;kx<=K1;kx++){
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){ for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
u[klookup_sym(kx,ky,K2)]+=tmp[klookup_sym(kx,ky,K2)]; u[klookup_sym(kx,ky,K2)]=tmp[klookup_sym(kx,ky,K2)];
} }
} }
// next delta to use in future steps (do not exceed max_delta) // next delta to use in future steps (do not exceed max_delta)
*next_delta=fmin(max_delta, (*delta)*pow(relative*tolerance/err,0.2)); *next_delta=fmin(max_delta, (*delta)*pow(tolerance/cost,0.2));
// keep enstrophy constant // keep enstrophy constant
if(keep_en_cst){ if(keep_en_cst){
@ -827,9 +784,9 @@ int ns_step_rkf45(
} }
// error too big: repeat with smaller step // error too big: repeat with smaller step
else{ else{
*delta=factor*(*delta)*pow(relative*tolerance/err,0.2); *delta=factor*(*delta)*pow(tolerance/cost,0.2);
// do not recompute k1 // 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,keep_en_cst,target_en,false); ns_step_rkf45(u,tolerance,factor,max_delta,adaptive_cost,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; return 0;
@ -843,7 +800,7 @@ int ns_step_rkbs32(
double tolerance, double tolerance,
double factor, double factor,
double max_delta, double max_delta,
unsigned int adaptive_norm, unsigned int adaptive_cost,
int K1, int K1,
int K2, int K2,
int N1, int N1,
@ -869,7 +826,7 @@ int ns_step_rkbs32(
bool compute_k1 bool compute_k1
){ ){
int kx,ky; int kx,ky;
double err,relative; double cost;
// k1: u(t) // k1: u(t)
// only compute it if it is the first step (otherwise, it has already been computed due to the First Same As Last property) // only compute it if it is the first step (otherwise, it has already been computed due to the First Same As Last property)
@ -894,7 +851,7 @@ int ns_step_rkbs32(
ns_rhs(k3, tmp, K1, K2, N1, N2, nu, L, g, fft1, fft2, ifft, irreversible); ns_rhs(k3, tmp, K1, K2, N1, N2, nu, L, g, fft1, fft2, ifft, irreversible);
// k4 : u(t+delta) // k4 : u(t+delta)
// tmp cpmputed here is the next step // tmp computed here is the next step
for(kx=0;kx<=K1;kx++){ for(kx=0;kx<=K1;kx++){
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){ for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
tmp[klookup_sym(kx,ky,K2)]=u[klookup_sym(kx,ky,K2)]+(*delta)*(2./9*(*k1)[klookup_sym(kx,ky,K2)]+1./3*k2[klookup_sym(kx,ky,K2)]+4./9*k3[klookup_sym(kx,ky,K2)]); tmp[klookup_sym(kx,ky,K2)]=u[klookup_sym(kx,ky,K2)]+(*delta)*(2./9*(*k1)[klookup_sym(kx,ky,K2)]+1./3*k2[klookup_sym(kx,ky,K2)]+4./9*k3[klookup_sym(kx,ky,K2)]);
@ -902,56 +859,19 @@ int ns_step_rkbs32(
} }
ns_rhs(*k4, tmp, K1, K2, N1, N2, nu, L, g, fft1, fft2, ifft, irreversible); ns_rhs(*k4, tmp, K1, K2, N1, N2, nu, L, g, fft1, fft2, ifft, irreversible);
// compute error // next step
err=0; for(kx=0;kx<=K1;kx++){
if(adaptive_norm==NORM_L1){ for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
relative=0; // U: store in k3, which is no longer needed
for(kx=0;kx<=K1;kx++){ k3[klookup_sym(kx,ky,K2)]=u[klookup_sym(kx,ky,K2)]+(*delta)*(7./24*(*k1)[klookup_sym(kx,ky,K2)]+1./4*k2[klookup_sym(kx,ky,K2)]+1./3*k3[klookup_sym(kx,ky,K2)]+1./8*(*k4)[klookup_sym(kx,ky,K2)]);
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
err+=cabs((*delta)*(5./72*(*k1)[klookup_sym(kx,ky,K2)]-1./12*k2[klookup_sym(kx,ky,K2)]-1./9*k3[klookup_sym(kx,ky,K2)]+1./8*(*k4)[klookup_sym(kx,ky,K2)]));
relative+=cabs(tmp[klookup_sym(kx,ky,K2)]-u[klookup_sym(kx,ky,K2)]);
}
} }
} }
else if(adaptive_norm==NORM_k3){
relative=0;
for(kx=0;kx<=K1;kx++){
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
err+=cabs((*delta)*(5./72*(*k1)[klookup_sym(kx,ky,K2)]-1./12*k2[klookup_sym(kx,ky,K2)]-1./9*k3[klookup_sym(kx,ky,K2)]+1./8*(*k4)[klookup_sym(kx,ky,K2)]))/pow(kx*kx+ky*ky,1.5);
relative+=cabs(tmp[klookup_sym(kx,ky,K2)]-u[klookup_sym(kx,ky,K2)])/pow(kx*kx+ky*ky,1.5);
}
}
}
else if(adaptive_norm==NORM_k32){
relative=0;
for(kx=0;kx<=K1;kx++){
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
err+=cabs((*delta)*(5./72*(*k1)[klookup_sym(kx,ky,K2)]-1./12*k2[klookup_sym(kx,ky,K2)]-1./9*k3[klookup_sym(kx,ky,K2)]+1./8*(*k4)[klookup_sym(kx,ky,K2)]))/pow(kx*kx+ky*ky,0.75);
relative+=cabs(tmp[klookup_sym(kx,ky,K2)]-u[klookup_sym(kx,ky,K2)])/pow(kx*kx+ky*ky,0.75);
}
}
}
else if(adaptive_norm==NORM_ENSTROPHY){
double sumu, sumU;
sumu=0;
sumU=0;
for(kx=0;kx<=K1;kx++){
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
err+=(kx*kx+ky*ky)*cabs2((*delta)*(5./72*(*k1)[klookup_sym(kx,ky,K2)]-1./12*k2[klookup_sym(kx,ky,K2)]-1./9*k3[klookup_sym(kx,ky,K2)]+1./8*(*k4)[klookup_sym(kx,ky,K2)]));
sumU+=(kx*kx+ky*ky)*cabs2(u[klookup_sym(kx,ky,K2)]+(*delta)*(7./24*(*k1)[klookup_sym(kx,ky,K2)]+1./4*k2[klookup_sym(kx,ky,K2)]+1./3*k3[klookup_sym(kx,ky,K2)]+1./8*(*k4)[klookup_sym(kx,ky,K2)]));
sumu+=(kx*kx+ky*ky)*cabs2(tmp[klookup_sym(kx,ky,K2)]);
}
}
err=sqrt(err);
relative=(sqrt(sumu)+sqrt(sumU))/sumu;
}
else{
fprintf(stderr,"bug: unknown norm: %u, contact ian.jauslin@rutgers,edu\n", adaptive_norm);
exit(-1);
}
// compare relative error with tolerance // compute cost
if(err<relative*tolerance){ cost=ns_adaptive_cost(tmp, k3, adaptive_cost, K1, K2, g, L);
// compare cost with tolerance
if(cost<tolerance){
// add to output // add to output
for(kx=0;kx<=K1;kx++){ for(kx=0;kx<=K1;kx++){
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){ for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
@ -959,7 +879,7 @@ int ns_step_rkbs32(
} }
} }
// next delta to use in future steps (do not exceed max_delta) // next delta to use in future steps (do not exceed max_delta)
*next_delta=fmin(max_delta, (*delta)*pow(relative*tolerance/err,1./3)); *next_delta=fmin(max_delta, (*delta)*pow(tolerance/cost,1./3));
// k1 in the next step will be this k4 (first same as last) // k1 in the next step will be this k4 (first same as last)
tmp=*k1; tmp=*k1;
@ -978,9 +898,9 @@ int ns_step_rkbs32(
} }
// error too big: repeat with smaller step // error too big: repeat with smaller step
else{ else{
*delta=factor*(*delta)*pow(relative*tolerance/err,1./3); *delta=factor*(*delta)*pow(tolerance/cost,1./3);
// this will reuse the same k1 without re-computing it // 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,keep_en_cst,target_en,false); ns_step_rkbs32(u,tolerance,factor,max_delta,adaptive_cost,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; return 0;
@ -993,7 +913,7 @@ int ns_step_rkdp54(
double tolerance, double tolerance,
double factor, double factor,
double max_delta, double max_delta,
unsigned int adaptive_norm, unsigned int adaptive_cost,
int K1, int K1,
int K2, int K2,
int N1, int N1,
@ -1021,7 +941,7 @@ int ns_step_rkdp54(
bool compute_k1 bool compute_k1
){ ){
int kx,ky; int kx,ky;
double err,relative; double cost;
// k1: u(t) // k1: u(t)
// only compute it if it is the first step (otherwise, it has already been computed due to the First Same As Last property) // only compute it if it is the first step (otherwise, it has already been computed due to the First Same As Last property)
@ -1079,56 +999,19 @@ int ns_step_rkdp54(
// store in k2, which is not needed anymore // store in k2, which is not needed anymore
ns_rhs(*k2, tmp, K1, K2, N1, N2, nu, L, g, fft1, fft2, ifft, irreversible); ns_rhs(*k2, tmp, K1, K2, N1, N2, nu, L, g, fft1, fft2, ifft, irreversible);
// compute error //next step
err=0; for(kx=0;kx<=K1;kx++){
if(adaptive_norm==NORM_L1){ for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
relative=0; // U: store in k6, which is not needed anymore
for(kx=0;kx<=K1;kx++){ k6[klookup_sym(kx,ky,K2)]=u[klookup_sym(kx,ky,K2)]+(*delta)*(5179./57600*(*k1)[klookup_sym(kx,ky,K2)]+7571./16695*k3[klookup_sym(kx,ky,K2)]+393./640*k4[klookup_sym(kx,ky,K2)]-92097./339200*k5[klookup_sym(kx,ky,K2)]+187./2100*k6[klookup_sym(kx,ky,K2)]+1./40*(*k2)[klookup_sym(kx,ky,K2)]);
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
err+=cabs((*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+=cabs(tmp[klookup_sym(kx,ky,K2)]-u[klookup_sym(kx,ky,K2)]);
}
} }
} }
else if(adaptive_norm==NORM_k3){
relative=0; // compute cost
for(kx=0;kx<=K1;kx++){ cost=ns_adaptive_cost(tmp, k6, adaptive_cost, K1, K2, g, L);
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
err+=cabs((*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)]))/pow(kx*kx+ky*ky,1.5);
relative+=cabs(tmp[klookup_sym(kx,ky,K2)]-u[klookup_sym(kx,ky,K2)])/pow(kx*kx+ky*ky,1.5);
}
}
}
else if(adaptive_norm==NORM_k32){
relative=0;
for(kx=0;kx<=K1;kx++){
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
err+=cabs((*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)]))/pow(kx*kx+ky*ky,0.75);
relative+=cabs(tmp[klookup_sym(kx,ky,K2)]-u[klookup_sym(kx,ky,K2)])/pow(kx*kx+ky*ky,0.75);
}
}
}
else if(adaptive_norm==NORM_ENSTROPHY){
double sumu, sumU;
sumu=0;
sumU=0;
for(kx=0;kx<=K1;kx++){
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
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)]));
sumU+=(kx*kx+ky*ky)*cabs2(u[klookup_sym(kx,ky,K2)]+(*delta)*(5179./57600*(*k1)[klookup_sym(kx,ky,K2)]+7571./16695*k3[klookup_sym(kx,ky,K2)]+393./640*k4[klookup_sym(kx,ky,K2)]-92097./339200*k5[klookup_sym(kx,ky,K2)]+187./2100*k6[klookup_sym(kx,ky,K2)]+1./40*(*k2)[klookup_sym(kx,ky,K2)]));
sumu+=(kx*kx+ky*ky)*cabs2(tmp[klookup_sym(kx,ky,K2)]);
}
}
err=sqrt(err);
relative=(sqrt(sumu)+sqrt(sumU))/sumu;
}
else{
fprintf(stderr,"bug: unknown norm: %u, contact ian.jauslin@rutgers,edu\n", adaptive_norm);
exit(-1);
}
// compare relative error with tolerance // compare relative error with tolerance
if(err<relative*tolerance){ if(cost<tolerance){
// add to output // add to output
for(kx=0;kx<=K1;kx++){ for(kx=0;kx<=K1;kx++){
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){ for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
@ -1136,9 +1019,9 @@ int ns_step_rkdp54(
} }
} }
// next delta to use in future steps (do not exceed max_delta) // next delta to use in future steps (do not exceed max_delta)
*next_delta=fmin(max_delta, (*delta)*pow(relative*tolerance/err,1./5)); *next_delta=fmin(max_delta, (*delta)*pow(tolerance/cost,0.2));
// k1 in the next step will be this k4 (first same as last) // k1 in the next step will be this k7 (first same as last)
tmp=*k1; tmp=*k1;
*k1=*k2; *k1=*k2;
*k2=tmp; *k2=tmp;
@ -1155,9 +1038,71 @@ int ns_step_rkdp54(
} }
// error too big: repeat with smaller step // error too big: repeat with smaller step
else{ else{
*delta=factor*(*delta)*pow(relative*tolerance/err,1./5); *delta=factor*(*delta)*pow(tolerance/cost,0.2);
// this will reuse the same k1 without re-computing it // 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,keep_en_cst,target_en,false); ns_step_rkdp54(u,tolerance,factor,max_delta,adaptive_cost,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;
}
// compute error for adaptive step methods
double ns_adaptive_cost(
_Complex double* u,
_Complex double* U,
unsigned int adaptive_cost,
int K1,
int K2,
_Complex double* g,
double L
){
int kx,ky;
if(adaptive_cost==COST_L1){
double cost=0;
double relative=0;
for(kx=0;kx<=K1;kx++){
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
cost+=cabs(u[klookup_sym(kx,ky,K2)]-U[klookup_sym(kx,ky,K2)]);
relative+=cabs(u[klookup_sym(kx,ky,K2)]);
}
}
return cost/relative;
}
else if(adaptive_cost==COST_k3){
double cost=0;
double relative=0;
for(kx=0;kx<=K1;kx++){
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
cost+=cabs(u[klookup_sym(kx,ky,K2)]-U[klookup_sym(kx,ky,K2)])/pow(kx*kx+ky*ky,3);
relative+=cabs(u[klookup_sym(kx,ky,K2)])/pow(kx*kx+ky*ky,3);
}
}
return cost/relative;
}
else if(adaptive_cost==COST_k32){
double cost=0;
double relative=0;
for(kx=0;kx<=K1;kx++){
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
cost+=cabs(u[klookup_sym(kx,ky,K2)]-U[klookup_sym(kx,ky,K2)])/pow(kx*kx+ky*ky,1.5);
relative+=cabs(u[klookup_sym(kx,ky,K2)])/pow(kx*kx+ky*ky,1.5);
}
}
return cost/relative;
}
else if(adaptive_cost==COST_ENSTROPHY){
double enu=compute_enstrophy(u,K1,K2,L);
return fabs(enu-compute_enstrophy(U,K1,K2,L))/enu;
}
else if(adaptive_cost==COST_ALPHA){
double alu=compute_alpha(u,K1,K2,g,L);
return fabs((alu-compute_alpha(U,K1,K2,g,L))/alu);
}
else{
fprintf(stderr,"bug: unknown norm: %u, contact ian.jauslin@rutgers.edu\n", adaptive_cost);
exit(-1);
} }
return 0; return 0;

18
src/navier-stokes.h
View File

@ -31,13 +31,13 @@ typedef struct fft_vects {
} fft_vect; } fft_vect;
// compute u_k // 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, bool keep_en_cst, double target_en, 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_cost, _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 // 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, bool keep_en_cst, double target_en, unsigned int algorithm, double print_freq, double starting_time, bool print_alpha, unsigned int nthreads, FILE* savefile, FILE* utfile, const char* cmd_string, const char* params_string, const char* savefile_string, const 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_cost, _Complex double* u0, _Complex double* g, bool irreversible, bool keep_en_cst, double target_en, unsigned int algorithm, double print_freq, double starting_time, bool print_alpha, unsigned int nthreads, FILE* savefile, FILE* utfile, const char* cmd_string, const char* params_string, const char* savefile_string, const char* utfile_string);
// compute solution as a function of time, but do not print anything (useful for debugging) // 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, bool keep_en_cst, double target_en, 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_cost, 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 // initialize vectors for computation
@ -49,17 +49,21 @@ 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); int copy_u( _Complex double* u, _Complex double* u0, int K1, int K2);
// next time step for Irreversible/reversible Navier-Stokes equation // 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); 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_cost, 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 // 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); 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 // 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, bool keep_en_cst, double target_en); 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 // 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 keep_en_cst, double target_en, bool compute_k1); int ns_step_rkf45( _Complex double* u, double tolerance, double factor, double max_delta, unsigned int adaptive_cost, 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 // 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 keep_en_cst, double target_en, bool compute_k1); int ns_step_rkdp54( _Complex double* u, double tolerance, double factor, double max_delta, unsigned int adaptive_cost, 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 // 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 keep_en_cst, double target_en, bool compute_k1); int ns_step_rkbs32( _Complex double* u, double tolerance, double factor, double max_delta, unsigned int adaptive_cost, 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);
// cost function for the adaptive iterations
double ns_adaptive_cost( _Complex double* u, _Complex double* U, unsigned int adaptive_cost, int K1, int K2, _Complex double* g, double L);
// right side of Irreversible/reversible Navier-Stokes equation // 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); 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);