-
Notifications
You must be signed in to change notification settings - Fork 0
/
ReactionNBurguers.py
307 lines (244 loc) · 9.82 KB
/
ReactionNBurguers.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
import numpy as np
from numpy.fft import fftn,ifftn,fftfreq
import math
import random
import itertools
import tqdm as tqdm
from collections import namedtuple
from scipy.signal import convolve2d
import cNoise
from scipy.interpolate import interp1d
class ReactionNBurguers():
'''
Wrote by: Rubens Andreas Sautter (2021)
Adapted from Huang, et.al.(2019)
https://advancesindifferenceequations.springeropen.com/articles/10.1186/s13662-019-2328-5
A Noise term can be applied to the advection term:
https://github.com/caseman/noise
Perlin Noise:
Lagae, Ares, et al. "A survey of procedural noise functions." Computer Graphics Forum. Vol. 29. No. 8. Oxford, UK: Blackwell Publishing Ltd, 2010.
Burguers-Bateman alternative equation solved with pseudospectral methods, and integrated with RK45.
du du d²u
__ + s W(t) u __ = v ___ + f(u,v)
dt dx dx²
dv dv d²v
__ + s W(t) v __ = v ___ + g(u,v)
dt dx dx²
'''
def __init__(self, alpha=1.13, b=0.1781, e1=0.3, e2=0.2, mu_v=1.0, mu_u=3.2,s_u=0.1, s_v=0.32, r=0.2,h=6.0, msize = 128,ic='r', mNoise='advection', kappa=0.1, noiseType=None,noiseArgs=None):
'''
Spatial parameters:
ic = initial condition('r', 'g')
h - grid spacing
mu_u - prey viscosity
mu_v - predator viscosity
s_v - predator spatial velocity - burguers and transport equation hybrid
s_u - prey spatial velocity - burguers and transport equation hybrid
gamma - reaction rate
Reaction parameters:
alpha - hunt rate
b - predator gain
e1 - inverse hunt rate
e2 - inverse predator gain
r - reaction speed
Noise Parameters:
mNoise - 'advection' - applies the noise on advection
- 'diffusion' - applies the noise on diffusion
kappa - noise velocity - [0-1]
noiseType - 'gradient' - Perlin noise (https://en.wikipedia.org/wiki/Perlin_noise)
- 'colored' - (https://en.wikipedia.org/wiki/Colors_of_noise)
- 'fixed' - gradient noise with the same values at each time step
- None - advection in a single direction
noiseArgs - Gradient noise - {'octaves':32,'persistence':0.5}
- Colored noise - {'beta':2}
- None - if none or any of the parameters is specified, the above parameters are applied
'''
self.alpha,self.b,self.e1,self.e2,self.r = alpha, b, e1, e2, r
self.mu_v, self.mu_u, self.s_u, self.s_v,self.h,self.ndim = mu_v, mu_u, s_u, s_v,h,2
u0,v0 = 0.5,0.6
u0 = 0.5*(1-alpha-e1+np.sqrt((alpha+e1-1)*(alpha+e1-1)-4*(alpha*e2-e1)))
v0 = u0+e2
print("Stable (u*,v*): ",u0,v0)
self.ic = ic
self.u0 ,self.v0 = u0,v0
self.msize = msize
self.dim = 2
self.noiseType=noiseType
self.mNoise =mNoise
self.kappa = kappa
if noiseArgs is None:
self.noiseArgs = {}
else:
self.noiseArgs = noiseArgs
def __getRandom(self,n,dim):
newShape = tuple([n for i in range(dim)])
return np.random.rand(n**dim).reshape(newShape)
def __getGaussian(self,n,dim):
out = np.zeros(np.repeat(n,dim))
c = n/2
squareDists = np.sum((np.indices(out.shape)-c)**2,as_vs=0)
return np.exp(-squareDists/n)
def getInitialCondition(self):
if self.ic=='r':
self.u = self.u0*(1+1e-2*(self.__getRandom(self.msize,self.dim)-0.5))
self.v = self.v0*(1+1e-2*(self.__getRandom(self.msize,self.dim)-0.5))
else:
self.u = self.u0*(1+1e-2*self.__getGaussian(self.msize,self.dim))
self.v = self.v0*(1+1e-2*self.__getGaussian(self.msize,self.dim))
return np.array([self.u, self.v])
def getChainedSingleReaction(self,u0=None,v0=None,dt=0.1, nit=3000):
ul,vl = [],[]
delta = 1e-6*(np.random.rand()-0.5)
if u0 is None:
ut = self.u0+delta
else:
ut = u0
if v0 is None:
vt = self.v0+delta
else:
vt = v0
for i in range(nit):
ul.append(ut)
vl.append(vt)
t = i*dt
state = np.array([ut,vt])
k1 = self.reaction(state[0],state[1],t)
k2 = self.reaction((state+dt*k1/2)[0],(state+dt*k1/2)[1], t+dt/2.)
k3 = self.reaction((state+dt*k2/2)[0],(state+dt*k2/2)[1], t+dt/2.)
k4 = self.reaction((state+dt*k3)[0],(state+dt*k3)[1], t+dt)
state = state + dt*(k1+2*k2+2*k3+k4)/6.
ut, vt = state
return ul, vl
def generatePNoise(self,nit,octaves=64,p=0.7):
pn = np.zeros((nit,self.msize,self.msize))
timeRand = np.random.randint(512)
for z, k in enumerate(tqdm.tqdm(np.linspace(0,1,nit))):
for x,i in enumerate(np.linspace(0,1,self.msize)):
for y, j in enumerate(np.linspace(0,1,self.msize)):
pn[z,x,y] = noise.snoise3(timeRand+k,i,j,octaves=octaves,persistence=p)
return np.array(pn)
def reaction(self,u,v,t):
alpha, e1, r = self.alpha, self.e1,self.r
b, e2, r = self.b, self.e2,self.r
ul,vl = u,v
u2 = r*(ul*(1-ul)-alpha*vl*ul/(ul+e1))
v2 = r*b*vl*(1-vl/(ul+e2))
return np.array([u2,v2])
def interpolateNoise(self,time):
'''
Linear interpolation of the noise
return the slice of nx and ny at the given time
'''
t = np.linspace(0,1,self.nx.shape[0])
p1, p2 = int(np.floor(self.nx.shape[0]*time)),int(np.ceil(self.nx.shape[0]*time))
t1 = t[p1]
t2 = t[p2]
mx1 = self.nx[p1]
my1 = self.ny[p1]
mx2 = self.nx[p2]
my2 = self.ny[p2]
if np.abs(t1-t2)<1e-15:
mx3,my3 = mx1,my1
else:
mx3 = np.abs(t1-time)*mx1/(np.abs(t2-t1))+np.abs(t2-time)*mx2/(np.abs(t2-t1))
my3 = np.abs(t1-time)*my1/(np.abs(t2-t1))+np.abs(t2-time)*my2/(np.abs(t2-t1))
return mx3, my3
def solveRKF45(self,dt,ntimes,stepsave,dtTolerace=1e-4):
state = self.getInitialCondition()
times = []
states = [self.getInitialCondition()]
w = np.array([ [ 0,0,0,0,0,0],
[1/4, 0,0,0,0,0],
[3/32,9/32, 0,0,0,0],
[1932/2197,-7200/2197,7296/2197, 0,0,0],
[439/216,-8,3680/513,-845/4104, 0,0],
[-8/27, 2,-3544/2565,1859/4104,-11/40,0]
])
t = 0.0
if self.noiseType == 'colored':
if 'beta' in self.noiseArgs:
exponent = self.noiseArgs['beta']
else:
exponent = 2
if 'std' in self.noiseArgs:
std = self.noiseArgs['std']
else:
std = 0.03
self.nx = cNoise.cNoise(beta=exponent,shape=(ntimes+2,self.msize,self.msize))
self.ny = cNoise.cNoise(beta=exponent,shape=(ntimes+2,self.msize,self.msize))
elif self.noiseType == 'gradient':
if 'octaves' in self.noiseArgs:
octaves = self.noiseArgs['octaves']
else:
octaves = 16
if 'persistence' in self.noiseArgs:
persistence = self.noiseArgs['persistence']
else:
persistence = 0.5
self.nx = self.generatePNoise(nit=ntimes+2,octaves=octaves,p=persistence)
self.ny = self.generatePNoise(nit=ntimes+2,octaves=octaves,p=persistence)
else:
self.nx = np.ones((ntimes+2,self.msize,self.msize))
self.ny = np.ones((ntimes+2,self.msize,self.msize))
self.maxTime = (ntimes+1)*dt
self.nx = self.nx/np.max(np.abs(self.nx[:int(self.kappa*self.maxTime)+1]))
self.ny = self.ny/np.max(np.abs(self.ny[:int(self.kappa*self.maxTime)+1]))
for time in tqdm.tqdm(range(ntimes)):
step = dt
k1 = step*self.spatialDerivatives(state, t )
k2 = step*self.spatialDerivatives(state+k1*w[1,0], t+step/4 )
k3 = step*self.spatialDerivatives(state+k1*w[2,0]+k2*w[2,1], t+3*step/8 )
k4 = step*self.spatialDerivatives(state+k1*w[3,0]+k2*w[3,1]+k3*w[3,2], t+12*step/13 )
k5 = step*self.spatialDerivatives(state+k1*w[4,0]+k2*w[4,1]+k3*w[4,2]+k4*w[4,3], t+step )
k6 = step*self.spatialDerivatives(state+k1*w[5,0]+k2*w[5,1]+k3*w[5,2]+k4*w[5,3]+k5*w[5,4], t+step/2 )
approach4 = state + (25/216)*k1 + (1408/2565)*k3 + (2197/4101)*k4 -k5/5
approach5 = state + (16/135)*k1 + (6656/12825)*k3 + (28561/56430)*k4 - (9/50)*k5 + (2/55)*k6
error = np.max(np.abs(approach4-approach5))
if error> dtTolerace:
step = dt*((dtTolerace/(2*error))**.25)
k1 = step*self.spatialDerivatives(state, t )
k2 = step*self.spatialDerivatives(state+k1*w[1,0], t+step/4 )
k3 = step*self.spatialDerivatives(state+k1*w[2,0]+k2*w[2,1], t+3*step/8 )
k4 = step*self.spatialDerivatives(state+k1*w[3,0]+k2*w[3,1]+k3*w[3,2], t+12*step/13 )
k5 = step*self.spatialDerivatives(state+k1*w[4,0]+k2*w[4,1]+k3*w[4,2]+k4*w[4,3], t+step )
k6 = step*self.spatialDerivatives(state+k1*w[5,0]+k2*w[5,1]+k3*w[5,2]+k4*w[5,3]+k5*w[5,4], t+step/2 )
approach4 = state + (25/216)*k1 + (1408/2565)*k3 + (2197/4101)*k4 -k5/5
t += step
state = approach4
state = state + step*self.reaction(state[0],state[1],t)
times.append(t)
if time in stepsave:
states.append(state)
return np.array(states), np.array(times)
def spatialDerivatives(self,state,time):
ul, vl = state
h, mu_u, mu_v, s_u, s_v = self.h,self.mu_u, self.mu_v, self.s_u, self.s_v
nx, ny = self.interpolateNoise(self.kappa*time/self.maxTime)
#PseudoSpectral approach:
fx = 2*np.pi*fftfreq(ul.shape[0])
fy = 2*np.pi*fftfreq(vl.shape[1])
ftu = fftn(ul)
ftv = fftn(vl)
lapU = np.real(ifftn( (-(fx[None,:]**2)-(fy[:,None]**2)) * ftu))
lapV = np.real(ifftn( (-(fx[None,:]**2)-(fy[:,None]**2)) * ftv))
derUx = np.real(ifftn( fx[None,:]*1j * ftu))
derUy = np.real(ifftn( fy[None,:]*1j * ftu))
derVx = np.real(ifftn( fx[None,:]*1j * ftv))
derVy = np.real(ifftn( fy[None,:]*1j * ftv))
# diffusion:
if self.mNoise == 'diffusion':
derUt = (mu_u/(h**2))*(np.power(nx,2)+1e-5)*lapU
derVt = (mu_v/(h**2))*(np.power(nx,2)+1e-5)*lapV
else:
derUt = (mu_u/(h**2))*lapU
derVt = (mu_v/(h**2))*lapV
# advection:
if self.mNoise == 'advection':
#derUt -= s_u*self.u*((nx)*derUx + (ny)*derUy)/h
#derVt -= s_v*self.v*((nx)*derVx + (ny)*derVy)/h
derUt -= s_u*self.u*((nx*np.cos(2*np.pi*ny))*derUx + (nx*np.sin(2*np.pi*ny))*derUy)/h
derVt -= s_v*self.v*((ny*np.cos(2*np.pi*nx))*derVx + (ny*np.cos(2*np.pi*nx))*derVy)/h
else:
derUt -= s_u*self.u*(derUx + derUy)/h
derVt -= s_v*self.v*(derVx + derVy)/h
return np.array([derUt, derVt])