Rewrite cost function for adaptive step
This commit is contained in:
parent
9fa10c8db4
commit
d81ad18618
@ -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
|
||||
exceet `delta_max`.
|
||||
|
||||
* `adaptive_norm`: norm to use to estimate the error of the adaptive method:
|
||||
`L1` (default) for the normalized L1 norm, `k3` for the normalized L1 norm of
|
||||
f_k/|k|^3, `k32` for the normalized L1 norm, `enstrophy` for the enstrophy
|
||||
norm.
|
||||
* `adaptive_cost`: cost function to use to estimate the error of the adaptive
|
||||
method: `L1` (default) for the normalized L1 norm, `k3` for the normalized L1
|
||||
norm of f_k/|k|^3, `k32` for the normalized L1 norm, `enstrophy` for the
|
||||
enstrophy, `alpha` for alpha.
|
||||
|
||||
* `keep_en_cst` (0 or 1, default 0): impose that the enstrophy is constant at
|
||||
each step (only really useful for the reversible equation).
|
||||
|
||||
@ -115,18 +115,18 @@ In addition, several variable step methods are implemented:
|
||||
\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)}\|
|
||||
D(\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.
|
||||
for some given {\it cost function} $D$, and $\epsilon_{\mathrm{target}}$, set using the {\tt adaptive\_tolerance} parameter.
|
||||
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.
|
||||
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}
|
||||
\|\hat u^{(n)}-\hat U^{(n)}\|=\delta_n^qC_n
|
||||
.
|
||||
D(\hat u^{(n)},\hat U^{(n)})=\delta_n^qC_n
|
||||
\end{equation}
|
||||
for some number $C_n$.
|
||||
We wish to set $\delta_{n+1}$ so that
|
||||
\begin{equation}
|
||||
\delta_{n+1}^qC_n=\epsilon_{\mathrm{target}}
|
||||
@ -135,7 +135,7 @@ 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}
|
||||
=\delta_n\left(\frac{\epsilon_{\mathrm{target}}}{D(\hat u^{(n)},\hat U^{(n)})}\right)^{\frac1q}
|
||||
.
|
||||
\label{adaptive_delta}
|
||||
\end{equation}
|
||||
@ -145,59 +145,49 @@ To be safe, we also set a maximal value for $\delta$ via the {\tt max\_delta} pa
|
||||
\bigskip
|
||||
|
||||
\indent
|
||||
The choice of the norm $\|\cdot\|$ matters.
|
||||
It can be made by specifying the parameter {\tt adaptive\_norm}.
|
||||
The choice of the cost function $D$ matters.
|
||||
It can be made by specifying the parameter {\tt adaptive\_cost}.
|
||||
\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}
|
||||
\|f\|:=
|
||||
\frac1{\mathcal N}\sum_k|f_k|
|
||||
,\quad
|
||||
\mathcal N:=\sum_k|\hat u_k^{(n)}-\hat u_k^{(n-1)}|
|
||||
D(\hat u,\hat U):=\frac{|\mathcal En(\hat u)-\mathcal En(\hat U)|}{\mathcal En(\hat u)}
|
||||
.
|
||||
\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}
|
||||
\|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
|
||||
\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}
|
||||
or
|
||||
\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
|
||||
\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}
|
||||
are sensible choices.
|
||||
These norms are selected by choosing {\tt adaptive\_norm=k3} and {\tt adaptive\_norm=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}.
|
||||
These cost functions are selected by choosing {\tt adaptive\_cost=k3} and {\tt adaptive\_cost=k32} respectively.
|
||||
\end{itemize}
|
||||
|
||||
\subsubsection{Reality}.
|
||||
@ -511,6 +501,7 @@ and so
|
||||
\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}
|
||||
.
|
||||
\label{alpha}
|
||||
\end{equation}
|
||||
Note that, by\-~(\ref{realu})-(\ref{realT}),
|
||||
\begin{equation}
|
||||
|
||||
@ -40,7 +40,8 @@ limitations under the License.
|
||||
#define ALGORITHM_RKDP54 1002
|
||||
#define ALGORITHM_RKBS32 1003
|
||||
|
||||
#define NORM_L1 1
|
||||
#define NORM_k3 2
|
||||
#define NORM_k32 3
|
||||
#define NORM_ENSTROPHY 4
|
||||
#define COST_L1 1
|
||||
#define COST_k3 2
|
||||
#define COST_k32 3
|
||||
#define COST_ENSTROPHY 4
|
||||
#define COST_ALPHA 5
|
||||
|
||||
48
src/main.c
48
src/main.c
@ -46,7 +46,7 @@ typedef struct nstrophy_parameters {
|
||||
double adaptive_tolerance;
|
||||
double adaptive_factor;
|
||||
double max_delta;
|
||||
unsigned int adaptive_norm;
|
||||
unsigned int adaptive_cost;
|
||||
double print_freq;
|
||||
int seed;
|
||||
double starting_time;
|
||||
@ -272,19 +272,19 @@ int main (
|
||||
|
||||
// run command
|
||||
if (command==COMMAND_UK){
|
||||
uk(parameters.K1, parameters.K2, parameters.N1, parameters.N2, parameters.final_time, parameters.nu, parameters.delta, parameters.L, parameters.adaptive_tolerance, parameters.adaptive_factor, parameters.max_delta, parameters.adaptive_norm, u0, g, parameters.irreversible, parameters.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){
|
||||
// register signal handler to handle aborts
|
||||
signal(SIGINT, sig_handler);
|
||||
signal(SIGTERM, sig_handler);
|
||||
enstrophy(parameters.K1, parameters.K2, parameters.N1, parameters.N2, parameters.final_time, parameters.nu, parameters.delta, parameters.L, parameters.adaptive_tolerance, parameters.adaptive_factor, parameters.max_delta, parameters.adaptive_norm, u0, g, parameters.irreversible, parameters.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(¶m_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(¶m_str), dstring_to_str_noinit(&savefile_str), dstring_to_str_noinit(&utfile_str));
|
||||
}
|
||||
else if(command==COMMAND_QUIET){
|
||||
quiet(parameters.K1, parameters.K2, parameters.N1, parameters.N2, parameters.final_time, parameters.nu, parameters.delta, parameters.L, parameters.adaptive_tolerance, parameters.adaptive_factor, parameters.max_delta, parameters.adaptive_norm, parameters.starting_time, u0, g, parameters.irreversible, parameters.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){
|
||||
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){
|
||||
fprintf(stderr, "error: no command specified\n");
|
||||
@ -393,18 +393,21 @@ int print_params(
|
||||
}
|
||||
|
||||
if(parameters.algorithm>ALGORITHM_ADAPTIVE_THRESHOLD){
|
||||
switch(parameters.adaptive_norm){
|
||||
case NORM_L1:
|
||||
fprintf(file,", norm=L1");
|
||||
switch(parameters.adaptive_cost){
|
||||
case COST_L1:
|
||||
fprintf(file,", cost=L1");
|
||||
break;
|
||||
case NORM_k3:
|
||||
fprintf(file,", norm=k3");
|
||||
case COST_k3:
|
||||
fprintf(file,", cost=k3");
|
||||
break;
|
||||
case NORM_k32:
|
||||
fprintf(file,", norm=k32");
|
||||
case COST_k32:
|
||||
fprintf(file,", cost=k32");
|
||||
break;
|
||||
case NORM_ENSTROPHY:
|
||||
fprintf(file,", norm=enstrophy");
|
||||
case COST_ENSTROPHY:
|
||||
fprintf(file,", cost=enstrophy");
|
||||
break;
|
||||
case COST_ALPHA:
|
||||
fprintf(file,", cost=alpha");
|
||||
break;
|
||||
}
|
||||
}
|
||||
@ -544,7 +547,7 @@ int set_default_params(
|
||||
parameters->adaptive_tolerance=1e-11;
|
||||
parameters->adaptive_factor=0.9;
|
||||
parameters->max_delta=1e-2;
|
||||
parameters->adaptive_norm=NORM_L1;
|
||||
parameters->adaptive_cost=COST_L1;
|
||||
parameters->final_time=100000;
|
||||
parameters->print_freq=1;
|
||||
parameters->starting_time=0;
|
||||
@ -759,21 +762,24 @@ int set_parameter(
|
||||
return(-1);
|
||||
}
|
||||
}
|
||||
else if (strcmp(lhs,"adaptive_norm")==0){
|
||||
else if (strcmp(lhs,"adaptive_cost")==0){
|
||||
if (strcmp(rhs,"L1")==0){
|
||||
parameters->adaptive_norm=NORM_L1;
|
||||
parameters->adaptive_cost=COST_L1;
|
||||
}
|
||||
else if (strcmp(rhs,"k3")==0){
|
||||
parameters->adaptive_norm=NORM_k3;
|
||||
parameters->adaptive_cost=COST_k3;
|
||||
}
|
||||
else if (strcmp(rhs,"k32")==0){
|
||||
parameters->adaptive_norm=NORM_k32;
|
||||
parameters->adaptive_cost=COST_k32;
|
||||
}
|
||||
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{
|
||||
fprintf(stderr, "error: unrecognized adaptive_norm '%s'\n",rhs);
|
||||
fprintf(stderr, "error: unrecognized adaptive_cost '%s'\n",rhs);
|
||||
return(-1);
|
||||
}
|
||||
}
|
||||
|
||||
@ -36,7 +36,7 @@ int uk(
|
||||
double adaptive_tolerance,
|
||||
double adaptive_factor,
|
||||
double max_delta,
|
||||
unsigned int adaptive_norm,
|
||||
unsigned int adaptive_cost,
|
||||
_Complex double* u0,
|
||||
_Complex double* g,
|
||||
bool irreversible,
|
||||
@ -90,7 +90,7 @@ int uk(
|
||||
time=starting_time;
|
||||
while(final_time<0 || time<final_time){
|
||||
// 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;
|
||||
step=next_step;
|
||||
@ -136,7 +136,7 @@ int enstrophy(
|
||||
double adaptive_tolerance,
|
||||
double adaptive_factor,
|
||||
double max_delta,
|
||||
unsigned int adaptive_norm,
|
||||
unsigned int adaptive_cost,
|
||||
_Complex double* u0,
|
||||
_Complex double* g,
|
||||
bool irreversible,
|
||||
@ -193,7 +193,7 @@ int enstrophy(
|
||||
// iterate
|
||||
time=starting_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;
|
||||
|
||||
@ -281,7 +281,7 @@ int quiet(
|
||||
double adaptive_tolerance,
|
||||
double adaptive_factor,
|
||||
double max_delta,
|
||||
unsigned int adaptive_norm,
|
||||
unsigned int adaptive_cost,
|
||||
double starting_time,
|
||||
_Complex double* u0,
|
||||
_Complex double* g,
|
||||
@ -317,7 +317,7 @@ int quiet(
|
||||
// iterate
|
||||
time=starting_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;
|
||||
step=next_step;
|
||||
@ -483,7 +483,7 @@ int ns_step(
|
||||
double adaptive_tolerance,
|
||||
double adaptive_factor,
|
||||
double max_delta,
|
||||
unsigned int adaptive_norm,
|
||||
unsigned int adaptive_cost,
|
||||
double L,
|
||||
_Complex double* g,
|
||||
double time,
|
||||
@ -508,15 +508,15 @@ int ns_step(
|
||||
} 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);
|
||||
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) {
|
||||
// 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);
|
||||
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) {
|
||||
// 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);
|
||||
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 {
|
||||
fprintf(stderr,"bug: unknown algorithm: %u, contact ian.jauslin@rutgers.edu\n",algorithm);
|
||||
}
|
||||
@ -674,7 +674,7 @@ int ns_step_rkf45(
|
||||
double tolerance,
|
||||
double factor,
|
||||
double max_delta,
|
||||
unsigned int adaptive_norm,
|
||||
unsigned int adaptive_cost,
|
||||
int K1,
|
||||
int K2,
|
||||
int N1,
|
||||
@ -701,7 +701,7 @@ int ns_step_rkf45(
|
||||
bool compute_k1
|
||||
){
|
||||
int kx,ky;
|
||||
double err,relative;
|
||||
double cost;
|
||||
|
||||
// k1: u(t)
|
||||
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);
|
||||
|
||||
// compute error
|
||||
err=0;
|
||||
if(adaptive_norm==NORM_L1){
|
||||
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)]));
|
||||
// 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)]);
|
||||
}
|
||||
// next step
|
||||
for(kx=0;kx<=K1;kx++){
|
||||
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
|
||||
// u
|
||||
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)]);
|
||||
// U: save to k6, which is no longer needed
|
||||
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)]);
|
||||
}
|
||||
}
|
||||
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)*(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);
|
||||
}
|
||||
|
||||
// cost function
|
||||
cost=ns_adaptive_cost(tmp,k6,adaptive_cost,K1,K2,g,L);
|
||||
|
||||
// compare relative error with tolerance
|
||||
if(err<relative*tolerance){
|
||||
// add to output
|
||||
if(cost<tolerance){
|
||||
// copy to output
|
||||
for(kx=0;kx<=K1;kx++){
|
||||
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=fmin(max_delta, (*delta)*pow(relative*tolerance/err,0.2));
|
||||
*next_delta=fmin(max_delta, (*delta)*pow(tolerance/cost,0.2));
|
||||
|
||||
// keep enstrophy constant
|
||||
if(keep_en_cst){
|
||||
@ -827,9 +784,9 @@ int ns_step_rkf45(
|
||||
}
|
||||
// error too big: repeat with smaller step
|
||||
else{
|
||||
*delta=factor*(*delta)*pow(relative*tolerance/err,0.2);
|
||||
*delta=factor*(*delta)*pow(tolerance/cost,0.2);
|
||||
// do not recompute k1
|
||||
ns_step_rkf45(u,tolerance,factor,max_delta,adaptive_norm,K1,K2,N1,N2,nu,delta,next_delta,L,g,fft1,fft2,ifft,k1,k2,k3,k4,k5,k6,tmp,irreversible,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;
|
||||
@ -843,7 +800,7 @@ int ns_step_rkbs32(
|
||||
double tolerance,
|
||||
double factor,
|
||||
double max_delta,
|
||||
unsigned int adaptive_norm,
|
||||
unsigned int adaptive_cost,
|
||||
int K1,
|
||||
int K2,
|
||||
int N1,
|
||||
@ -869,7 +826,7 @@ int ns_step_rkbs32(
|
||||
bool compute_k1
|
||||
){
|
||||
int kx,ky;
|
||||
double err,relative;
|
||||
double cost;
|
||||
|
||||
// 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)
|
||||
@ -894,7 +851,7 @@ int ns_step_rkbs32(
|
||||
ns_rhs(k3, tmp, K1, K2, N1, N2, nu, L, g, fft1, fft2, ifft, irreversible);
|
||||
|
||||
// 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(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)]);
|
||||
@ -902,56 +859,19 @@ int ns_step_rkbs32(
|
||||
}
|
||||
ns_rhs(*k4, tmp, K1, K2, N1, N2, nu, L, g, fft1, fft2, ifft, irreversible);
|
||||
|
||||
// compute error
|
||||
err=0;
|
||||
if(adaptive_norm==NORM_L1){
|
||||
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)]));
|
||||
relative+=cabs(tmp[klookup_sym(kx,ky,K2)]-u[klookup_sym(kx,ky,K2)]);
|
||||
}
|
||||
// next step
|
||||
for(kx=0;kx<=K1;kx++){
|
||||
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
|
||||
// U: store in k3, which is no longer needed
|
||||
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)]);
|
||||
}
|
||||
}
|
||||
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
|
||||
if(err<relative*tolerance){
|
||||
// compute cost
|
||||
cost=ns_adaptive_cost(tmp, k3, adaptive_cost, K1, K2, g, L);
|
||||
|
||||
// compare cost with tolerance
|
||||
if(cost<tolerance){
|
||||
// add to output
|
||||
for(kx=0;kx<=K1;kx++){
|
||||
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=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)
|
||||
tmp=*k1;
|
||||
@ -978,9 +898,9 @@ int ns_step_rkbs32(
|
||||
}
|
||||
// error too big: repeat with smaller step
|
||||
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
|
||||
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;
|
||||
@ -993,7 +913,7 @@ int ns_step_rkdp54(
|
||||
double tolerance,
|
||||
double factor,
|
||||
double max_delta,
|
||||
unsigned int adaptive_norm,
|
||||
unsigned int adaptive_cost,
|
||||
int K1,
|
||||
int K2,
|
||||
int N1,
|
||||
@ -1021,7 +941,7 @@ int ns_step_rkdp54(
|
||||
bool compute_k1
|
||||
){
|
||||
int kx,ky;
|
||||
double err,relative;
|
||||
double cost;
|
||||
|
||||
// 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)
|
||||
@ -1079,56 +999,19 @@ int ns_step_rkdp54(
|
||||
// store in k2, which is not needed anymore
|
||||
ns_rhs(*k2, tmp, K1, K2, N1, N2, nu, L, g, fft1, fft2, ifft, irreversible);
|
||||
|
||||
// compute error
|
||||
err=0;
|
||||
if(adaptive_norm==NORM_L1){
|
||||
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)]));
|
||||
relative+=cabs(tmp[klookup_sym(kx,ky,K2)]-u[klookup_sym(kx,ky,K2)]);
|
||||
}
|
||||
//next step
|
||||
for(kx=0;kx<=K1;kx++){
|
||||
for(ky=(kx>0 ? -K2 : 1);ky<=K2;ky++){
|
||||
// U: store in k6, which is not needed anymore
|
||||
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)]);
|
||||
}
|
||||
}
|
||||
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)*(-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);
|
||||
}
|
||||
|
||||
// compute cost
|
||||
cost=ns_adaptive_cost(tmp, k6, adaptive_cost, K1, K2, g, L);
|
||||
|
||||
// compare relative error with tolerance
|
||||
if(err<relative*tolerance){
|
||||
if(cost<tolerance){
|
||||
// add to output
|
||||
for(kx=0;kx<=K1;kx++){
|
||||
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=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;
|
||||
*k1=*k2;
|
||||
*k2=tmp;
|
||||
@ -1155,9 +1038,71 @@ int ns_step_rkdp54(
|
||||
}
|
||||
// error too big: repeat with smaller step
|
||||
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
|
||||
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;
|
||||
|
||||
@ -31,13 +31,13 @@ typedef struct fft_vects {
|
||||
} fft_vect;
|
||||
|
||||
// compute u_k
|
||||
int uk( int K1, int K2, int N1, int N2, double final_time, double nu, double delta, double L, double adaptive_tolerance, double adaptive_factor, double max_delta, unsigned int adaptive_norm, _Complex double* u0, _Complex double* g, bool irreversible, 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
|
||||
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)
|
||||
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
|
||||
@ -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);
|
||||
|
||||
// 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
|
||||
int ns_step_rk4( _Complex double* u, int K1, int K2, int N1, int N2, double nu, double delta, double L, _Complex double* g, fft_vect fft1, fft_vect fft2,fft_vect ifft, _Complex double* tmp1, _Complex double *tmp2, _Complex double *tmp3, bool irreversible, bool keep_en_cst, double target_en);
|
||||
// RK2
|
||||
int ns_step_rk2( _Complex double* u, int K1, int K2, int N1, int N2, double nu, double delta, double L, _Complex double* g, fft_vect fft1, fft_vect fft2,fft_vect ifft, _Complex double* tmp1, _Complex double *tmp2, bool irreversible, bool keep_en_cst, double target_en);
|
||||
// Runge-Kutta-Fehlberg
|
||||
int ns_step_rkf45( _Complex double* u, double tolerance, double factor, double max_delta, unsigned int adaptive_norm, int K1, int K2, int N1, int N2, double nu, double* delta, double* next_delta, double L, _Complex double* g, fft_vect fft1, fft_vect fft2, fft_vect ifft, _Complex double* k1, _Complex double* k2, _Complex double* k3, _Complex double* k4, _Complex double* k5, _Complex double* k6, _Complex double* tmp, bool irreversible, bool 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
|
||||
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
|
||||
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
|
||||
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);
|
||||
|
||||
Loading…
Reference in New Issue
Block a user