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Simpler expression for fft term

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This commit is contained in:
Ian Jauslin
2018-01-12 19:20:59 +00:00
parent 7ee5507b93
commit cff1d2ee3c
3 changed files with 57 additions and 68 deletions
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45
docs/nstrophy_doc/nstrophy_doc.tex
View File

@@ -49,16 +49,23 @@ in terms of which, multiplying both sides of the equation by $\frac{k^\perp}{|k|
\begin{equation}
\partial_t\hat \varphi_k=
-4\pi^2\nu k^2\hat \varphi_k+\frac{k^\perp}{2i\pi|k|}\cdot\hat g_k
-\frac{2i\pi}{|k|}\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}}
+\frac{4\pi^2}{|k|}\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}}
(q\cdot p^\perp)(k^\perp\cdot q^\perp)\hat\varphi_p\hat\varphi_q
.
\label{ins_k}
\end{equation}
which, since $q\cdot p^\perp=q\cdot(k^\perp-q^\perp)=q\cdot k^\perp$, is
Furthermore
\begin{equation}
(q\cdot p^\perp)(k^\perp\cdot q^\perp)
=
(q\cdot p^\perp)(q^2+p\cdot q)
\end{equation}
and $q\cdot p^\perp$ is antisymmetric under exchange of $q$ and $p$. Therefore,
\begin{equation}
\partial_t\hat \varphi_k=
-4\pi^2\nu k^2\hat \varphi_k+\frac{k^\perp}{2i\pi|k|}\cdot\hat g_k
+\frac{4\pi^2}{|k|}\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}}
(q\cdot k^\perp)(k\cdot q)\frac{\hat\varphi_p\hat\varphi_q}{|p||q|}
\frac{(q\cdot p^\perp)|q|}{|p|}\hat\varphi_p\hat\varphi_q
.
\label{ins_k}
\end{equation}
@@ -73,8 +80,7 @@ We truncate the Fourier modes and assume that $\hat\varphi_k=0$ if $|k_1|>K_1$ o
\begin{equation}
T(\hat\varphi,k):=
\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}}
\frac{\hat\varphi_p}{|p|}
(q\cdot k^\perp)(k\cdot q)\frac{\hat\varphi_q}{|q|}
\frac{(q\cdot p^\perp)|q|}{|p|}\hat\varphi_q\hat\varphi_p
\end{equation}
using a fast Fourier transform, defined as
\begin{equation}
@@ -98,8 +104,7 @@ in which $\mathcal F^*$ is defined like $\mathcal F$ but with the opposite phase
\sum_{p,q\in\mathcal K}
\frac1{N_1N_2}
\sum_{n\in\mathcal N}e^{-\frac{2i\pi}{N_1}n_1(p_1+q_1-k_1)-\frac{2i\pi}{N_2}n_2(p_2+q_2-k_2)}
\frac{\hat\varphi_p}{|p|}
(q\cdot k^\perp)(k\cdot q)\frac{\hat\varphi_q}{|q|}
(q\cdot p^\perp)\frac{|q|}{|p|}\hat\varphi_q\hat\varphi_p
\end{equation}
provided
\begin{equation}
@@ -109,29 +114,17 @@ Indeed, $\sum_{n_i=0}^{N_i}e^{-\frac{2i\pi}{N_i}n_im_i}$ vanishes unless $m_i=0\
\begin{equation}
T(\hat\varphi,k)
=
\textstyle
\frac1{N_1N_2}
\mathcal F^*\left(
\mathcal F(|p|^{-1}\hat\varphi_p)(n)
\mathcal F((|q|^{-1}(q\cdot k^\perp)(k\cdot q)\hat\varphi_q)_q)(n)
\right)(k)
\end{equation}
which we expand
\begin{equation}
T(\hat\varphi,k)
=
\textstyle
\frac{k_x^2-k_y^2}{N_1N_2}
\mathcal F^*\left(
\mathcal F\left(\frac{\hat\varphi_p}{|p|}\right)(n)
\mathcal F\left(\frac{q_xq_y}{|q|}\hat\varphi_q\right)(n)
\right)(k)
-
\frac{k_xk_y}{N_1N_2}
\mathcal F^*\left(
\mathcal F\left(\frac{\hat\varphi_p}{|p|}\right)(n)
\mathcal F\left(\frac{q_x^2-q_y^2}{|q|}\hat\varphi_q\right)(n)
\mathcal F\left(\frac{p_x\hat\varphi_p}{|p|}\right)(n)
\mathcal F\left(q_y|q|\hat\varphi_q\right)(n)
-
\mathcal F\left(\frac{p_y\hat\varphi_p}{|p|}\right)(n)
\mathcal F\left(q_x|q|\hat\varphi_q\right)(n)
\right)(k)
\end{equation}
\bigskip
\vfill

10
src/main.c
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@@ -77,10 +77,10 @@ int read_args(int argc, const char* argv[], ns_params* params, unsigned int* nst
*nsteps=16777216;
params->nu=4.9632717887631524e-05;
*/
params->h=1e-5;
params->K=16;
params->h=1e-3/(2*params->K+1);
*nsteps=10000000;
params->nu=1e-4;
params->nu=1./1024/(2*params->K+1);
// loop over arguments
for(i=1;i<argc;i++){
@@ -221,7 +221,7 @@ int enstrophy(ns_params params, unsigned int Nsteps){
for(ky=-params.K;ky<=params.K;ky++){
//params.g[KLOOKUP(kx,ky,params.S)]=sqrt(kx*kx*ky*ky)*exp(-(kx*kx+ky*ky));
if((kx==2 && ky==-1) || (kx==-2 && ky==1)){
params.g[KLOOKUP(kx,ky,params.S)]=1;
params.g[KLOOKUP(kx,ky,params.S)]=1.0*params.K;
}
else{
params.g[KLOOKUP(kx,ky,params.S)]=0;
@@ -246,19 +246,21 @@ int enstrophy(ns_params params, unsigned int Nsteps){
ins_step(u, params, fft_vects, tmp1, tmp2, tmp3);
alpha=compute_alpha(u, params);
/*
// to avoid errors building up in imaginary part
for(kx=-params.K;kx<=params.K;kx++){
for(ky=-params.K;ky<=params.K;ky++){
u[KLOOKUP(kx,ky,params.S)]=__real__ u[KLOOKUP(kx,ky,params.S)];
}
}
*/
// running average
if(t>0){
avg=avg-(avg-alpha)/t;
}
if(t>0 && t%1000==0){
if(t>0 && t%1==0){
fprintf(stderr,"%8d % .8e % .8e % .8e % .8e\n",t, __real__ avg, __imag__ avg, __real__ alpha, __imag__ alpha);
printf("%8d % .8e % .8e % .8e % .8e\n",t, __real__ avg, __imag__ avg, __real__ alpha, __imag__ alpha);
}

70
src/navier-stokes.c
View File

@@ -68,7 +68,7 @@ int ins_step(_Complex double* u, ns_params params, fft_vects vects, _Complex dou
int ins_rhs(_Complex double* out, _Complex double* u, ns_params params, fft_vects vects){
int kx,ky;
// F(u/|p|)*F(q1*q2*u/|q|)
// F(px/|p|*u)*F(qy*|q|*u)
// init to 0
for(kx=0; kx<params.N*params.N; kx++){
vects.fft1[kx]=0;
@@ -79,11 +79,12 @@ int ins_rhs(_Complex double* out, _Complex double* u, ns_params params, fft_vect
for(kx=-params.K;kx<=params.K;kx++){
for(ky=-params.K;ky<=params.K;ky++){
if(kx!=0 || ky!=0){
vects.fft1[KLOOKUP(kx,ky,params.N)]=u[KLOOKUP(kx,ky,params.S)]/sqrt(kx*kx+ky*ky);
vects.fft2[KLOOKUP(kx,ky,params.N)]=kx*ky*u[KLOOKUP(kx,ky,params.S)]/sqrt(kx*kx+ky*ky);
vects.fft1[KLOOKUP(kx,ky,params.N)]=kx/sqrt(kx*kx+ky*ky)*u[KLOOKUP(kx,ky,params.S)];
vects.fft2[KLOOKUP(kx,ky,params.N)]=ky*sqrt(kx*kx+ky*ky)*u[KLOOKUP(kx,ky,params.S)];
}
}
}
// fft
fftw_execute(vects.fft1_plan);
fftw_execute(vects.fft2_plan);
@@ -93,6 +94,33 @@ int ins_rhs(_Complex double* out, _Complex double* u, ns_params params, fft_vect
vects.invfft[KLOOKUP(kx,ky,params.N)]=vects.fft1[KLOOKUP(kx,ky,params.N)]*vects.fft2[KLOOKUP(kx,ky,params.N)];
}
}
// F(py/|p|*u)*F(qx*|q|*u)
// init to 0
for(kx=0; kx<params.N*params.N; kx++){
vects.fft1[kx]=0;
vects.fft2[kx]=0;
}
// fill modes
for(kx=-params.K;kx<=params.K;kx++){
for(ky=-params.K;ky<=params.K;ky++){
if(kx!=0 || ky!=0){
vects.fft1[KLOOKUP(kx,ky,params.N)]=ky/sqrt(kx*kx+ky*ky)*u[KLOOKUP(kx,ky,params.S)];
vects.fft2[KLOOKUP(kx,ky,params.N)]=kx*sqrt(kx*kx+ky*ky)*u[KLOOKUP(kx,ky,params.S)];
}
}
}
// fft
fftw_execute(vects.fft1_plan);
fftw_execute(vects.fft2_plan);
// write to invfft
for(kx=-2*params.K;kx<=2*params.K;kx++){
for(ky=-2*params.K;ky<=2*params.K;ky++){
vects.invfft[KLOOKUP(kx,ky,params.N)]=vects.invfft[KLOOKUP(kx,ky,params.N)]-vects.fft1[KLOOKUP(kx,ky,params.N)]*vects.fft2[KLOOKUP(kx,ky,params.N)];
}
}
// inverse fft
fftw_execute(vects.invfft_plan);
@@ -103,41 +131,7 @@ int ins_rhs(_Complex double* out, _Complex double* u, ns_params params, fft_vect
for(kx=-params.K;kx<=params.K;kx++){
for(ky=-params.K;ky<=params.K;ky++){
if(kx!=0 || ky!=0){
out[KLOOKUP(kx,ky,params.S)]=-4*M_PI*M_PI*params.nu*(kx*kx+ky*ky)*u[KLOOKUP(kx,ky,params.S)]+params.g[KLOOKUP(kx,ky,params.S)]+4*M_PI*M_PI/sqrt(kx*kx+ky*ky)*vects.invfft[KLOOKUP(kx,ky,params.N)]*(kx*kx-ky*ky)/params.N/params.N;
}
}
}
// F(u/|p|)*F((q1*q1-q2*q2)*u/|q|)
// init to 0
for(kx=0; kx<params.N*params.N; kx++){
vects.fft2[kx]=0;
vects.invfft[kx]=0;
}
// fill modes
for(kx=-params.K;kx<=params.K;kx++){
for(ky=-params.K;ky<=params.K;ky++){
if(kx!=0 || ky!=0){
vects.fft2[KLOOKUP(kx,ky,params.N)]=(kx*kx-ky*ky)*u[KLOOKUP(kx,ky,params.S)]/sqrt(kx*kx+ky*ky);
}
}
}
// fft
fftw_execute(vects.fft2_plan);
// write to invfft
for(kx=-2*params.K;kx<=2*params.K;kx++){
for(ky=-2*params.K;ky<=2*params.K;ky++){
vects.invfft[KLOOKUP(kx,ky,params.N)]=vects.fft1[KLOOKUP(kx,ky,params.N)]*vects.fft2[KLOOKUP(kx,ky,params.N)];
}
}
// inverse fft
fftw_execute(vects.invfft_plan);
// write out
for(kx=-params.K;kx<=params.K;kx++){
for(ky=-params.K;ky<=params.K;ky++){
if(kx!=0 || ky!=0){
out[KLOOKUP(kx,ky,params.S)]=out[KLOOKUP(kx,ky,params.S)]-4*M_PI*M_PI/sqrt(kx*kx+ky*ky)*vects.invfft[KLOOKUP(kx,ky,params.N)]*(kx*ky)/params.N/params.N;
out[KLOOKUP(kx,ky,params.S)]=-4*M_PI*M_PI*params.nu*(kx*kx+ky*ky)*u[KLOOKUP(kx,ky,params.S)]+params.g[KLOOKUP(kx,ky,params.S)]+4*M_PI*M_PI/sqrt(kx*kx+ky*ky)*vects.invfft[KLOOKUP(kx,ky,params.N)]/params.N/params.N;
}
}
}