Gray-Scott Model#
This notebook is originally an idea of @gouarin
import sys
import numpy as np
The reaction-diffusion system described here involves two generic chemical species U and V, whose concentration at a given point in space is referred to by variables u and v. As the term implies, they react with each other, and they diffuse through the medium. Therefore the concentration of U and V at any given location changes with time and can differ from that at other locations.
The overall behavior of the system is described by the following formula, two equations which describe three sources of increase and decrease for each of the two chemicals:
\[\begin{split}
\begin{array}{l}
\displaystyle \frac{\partial u}{\partial t} = D_u \Delta u - uv^2 + F(1-u) \\
\displaystyle \frac{\partial v}{\partial t} = D_v \Delta v + uv^2 - (F+k)v
\end{array}
\end{split}\]
The laplacian is computed with the following numerical scheme
\[
\Delta u_{i,j} \approx u_{i,j-1} + u_{i-1,j} -4u_{i,j} + u_{i+1, j} + u_{i, j+1}
\]
The classic Euler scheme is used to integrate the time derivative.
Initialization#
\(u\) is \(1\) everywhere et \(v\) is \(0\) in the domain except in a square zone where \(v = 0.25\) and \( u = 0.5\). This square located in the center of the domain is \([0, 1]\times[0,1]\) with a size of \(0.2\).
def init(n):
u = np.ones((n+2,n+2))
v = np.zeros((n+2,n+2))
x, y = np.meshgrid(np.linspace(0, 1, n+2), np.linspace(0, 1, n+2))
mask = (0.4<x) & (x<0.6) & (0.4<y) & (y<0.6)
u[mask] = 0.50
v[mask] = 0.25
return u, v
Boundary conditions#
We assume that the domain is periodic.
def periodic_bc(u):
u[0, :] = u[-2, :]
u[-1, :] = u[1, :]
u[:, 0] = u[:, -2]
u[:, -1] = u[:, 1]
Laplacian#
def laplacian(u):
"""
second order finite differences
"""
return ( u[ :-2, 1:-1] +
u[1:-1, :-2] - 4*u[1:-1, 1:-1] + u[1:-1, 2:] +
+ u[2: , 1:-1] )
Gray-Scott model#
def numpy_grayscott(U, V, Du, Dv, F, k):
u, v = U[1:-1,1:-1], V[1:-1,1:-1]
Lu = laplacian(U)
Lv = laplacian(V)
uvv = u*v*v
u += Du*Lu - uvv + F*(1 - u)
v += Dv*Lv + uvv - (F + k)*v
periodic_bc(U)
periodic_bc(V)
return U, V
Visualization#
Nous utiliserons les données suivantes.
Du, Dv = .1, .05
F, k = 0.0545, 0.062
%%time
from tqdm.notebook import tqdm
from PIL import Image
U, V = init(300)
def create_image(grayscott):
global U, V
for t in range(40):
U, V = grayscott(U, V, Du, Dv, F, k)
V_scaled = np.uint8(255*(V-V.min()) / (V.max()-V.min()))
return V_scaled
def create_frames(n, grayscott):
return [create_image(grayscott) for i in tqdm(range(n))]
frames = create_frames(500, numpy_grayscott)
CPU times: user 28.5 s, sys: 20.6 s, total: 49.1 s
Wall time: 48.9 s
from ipywidgets import interact, IntSlider
def display_sequence(iframe):
return Image.fromarray(frames[iframe])
interact(display_sequence,
iframe=IntSlider(min=0,
max=len(frames)-1,
step=1,
value=0,
continuous_update=True))
<function __main__.display_sequence(iframe)>
import imageio
frames_scaled = [np.uint8(255 * frame) for frame in frames]
imageio.mimsave('movie.gif', frames_scaled, format='gif', fps=60)
References#
Cython#
%load_ext cython
%%cython
cimport cython
import numpy as np
cimport numpy as np
cpdef cython_grayscott(np.ndarray[double, ndim=2] U, np.ndarray[double, ndim=2] V, double Du, double Dv, double F, double k):
cdef np.ndarray u = U[1:-1,1:-1]
cdef np.ndarray v = V[1:-1,1:-1]
cdef np.ndarray Lu = np.zeros_like(u)
cdef np.ndarray Lv = np.zeros_like(v)
cdef Py_ssize_t i, c, r, r1, c1, r2, c2
cdef double uvv
cdef double[:, ::1] bU = U
cdef double[:, ::1] bV = V
cdef double[:, ::1] bLu = Lu
cdef double[:, ::1] bLv = Lv
n = np.shape(U)[0]-2
for r in range(n):
r1 = r + 1
r2 = r + 2
for c in range(n):
c1 = c + 1
c2 = c + 2
bLu[r,c] = bU[r1,c2] + bU[r1,c] + bU[r2,c1] + bU[r,c1] - 4*bU[r1,c1]
bLv[r,c] = bV[r1,c2] + bV[r1,c] + bV[r2,c1] + bV[r,c1] - 4*bV[r1,c1]
for r in range(n):
r1 = r + 1
for c in range(n):
c1 = c + 1
uvv = bU[r1,c1]*bV[r1,c1]*bV[r1,c1]
bU[r1,c1] += Du*bLu[r,c] - uvv + F*(1 - bU[r1,c1])
bV[r1,c1] += Dv*bLv[r,c] + uvv - (F + k)*bV[r1,c1]
return U, V
Content of stderr:
In file included from /usr/share/miniconda/envs/python-fortran/lib/python3.9/site-packages/numpy/_core/include/numpy/ndarraytypes.h:1909,
from /usr/share/miniconda/envs/python-fortran/lib/python3.9/site-packages/numpy/_core/include/numpy/ndarrayobject.h:12,
from /usr/share/miniconda/envs/python-fortran/lib/python3.9/site-packages/numpy/_core/include/numpy/arrayobject.h:5,
from /home/runner/.cache/ipython/cython/_cython_magic_ac1ec9e1e79f03bdb61ccb46ffc03be560447f41.c:1250:
/usr/share/miniconda/envs/python-fortran/lib/python3.9/site-packages/numpy/_core/include/numpy/npy_1_7_deprecated_api.h:17:2: warning: #warning "Using deprecated NumPy API, disable it with " "#define NPY_NO_DEPRECATED_API NPY_1_7_API_VERSION" [-Wcpp]
17 | #warning "Using deprecated NumPy API, disable it with " \
| ^~~~~~~
U, V = init(300)
frames = create_frames(500, cython_grayscott)
interact(display_sequence,
iframe=IntSlider(min=0,
max=len(frames)-1,
step=1,
value=0,
continuous_update=True))
<function __main__.display_sequence(iframe)>
%load_ext fortranmagic
%%fortran
subroutine fortran_grayscott( U, V, Du, Dv, F, k)
real(8), intent(in) :: Du, Dv, F, k
real(8) :: U(:,:)
real(8) :: V(:,:)
real(8), allocatable :: Lu(:,:), Lv(:,:)
integer :: n, c, r, r1, c1, r2, c2
real(8) :: uvv
n = size(U,1)
Lu = U(2:n-1,2:n-1)
Lv = V(2:n-1,2:n-1)
do c = 1, n-2
c1 = c + 1
c2 = c + 2
do r = 1, n-2
r1 = r + 1
r2 = r + 2
Lu(r,c) = U(r1,c2) + U(r1,c) + U(r2,c1) + U(r,c1) - 4*U(r1,c1)
Lv(r,c) = V(r1,c2) + V(r1,c) + V(r2,c1) + V(r,c1) - 4*V(r1,c1)
end do
end do
do c = 1, n-2
c1 = c + 1
do r = 1, n-2
r1 = r + 1
uvv = U(r1,c1)*V(r1,c1)*V(r1,c1)
U(r1,c1) = U(r1, c1) + Du*Lu(r,c) - uvv + F*(1 - U(r1,c1))
V(r1,c1) = V(r1, c1) + Dv*Lv(r,c) + uvv - (F + k)*V(r1,c1)
end do
end do
end subroutine fortran_grayscott
U, V = init(300)
U = np.asfortranarray(U)
V = np.asfortranarray(V)
def create_image():
global U, V
for t in range(40):
fortran_grayscott(U, V, Du, Dv, F, k)
V_scaled = np.uint8(255*(V-V.min()) / (V.max()-V.min()))
return V_scaled
def create_frames(n):
return [create_image() for i in tqdm(range(n))]
frames = create_frames(500)
interact(display_sequence,
iframe=IntSlider(min=0,
max=len(frames)-1,
step=1,
value=0,
continuous_update=True))
<function __main__.display_sequence(iframe)>
U, V = init(300)
%timeit numpy_grayscott(U, V, Du, Dv, F, k)
1.27 ms ± 4.04 µs per loop (mean ± std. dev. of 7 runs, 1,000 loops each)
U, V = init(300)
%timeit cython_grayscott(U, V, Du, Dv, F, k)
780 µs ± 680 ns per loop (mean ± std. dev. of 7 runs, 1,000 loops each)
U, V = init(300)
U = np.asfortranarray(U)
V = np.asfortranarray(V)
%timeit fortran_grayscott(U, V, Du, Dv, F, k)
214 µs ± 1.32 µs per loop (mean ± std. dev. of 7 runs, 1,000 loops each)