def A_local_model_balflux(x,lamda,s,p):
"""
Balanced flux Evans matrix for the system named local_model.
"""
# wave speed
spd = p['spd']
# parameters
v_neg = p['v_neg']
tau_neg = p['tau_neg']
S_neg = p['S_neg']
m1 = p['m1']
none = p['none']
mu = p['mu']
kappa = p['kappa']
# interpolate profile solution
temp = soln(x,s)
#profile values
tau = temp[0]
S = temp[1]
ynot = np.array([tau,S])
temp2 = F_local_model(x,ynot,s,p);
tau_x = temp2[0]
S_x = temp2[1]
v = v_neg - spd * (tau - tau_neg)
v_x = -spd * tau_x
T_x = S_x * np.exp(S) / tau - np.exp(S) / tau ** 2 * tau_x
# Evans matrix
out = np.array([
[ lamda / spd, 0, 0, -0.1e1 / spd, 0 ],
[ 0, 0, 0, 1, 0 ],
[ lamda * (-np.exp(S) / tau ** 2 + tau - (np.exp(S) / tau + 0.1e1)
/ tau) / spd, 0, 0, -(-np.exp(S) / tau ** 2 + tau - (np.exp(S)
/ tau + 0.1e1) / tau) / spd + v, (np.exp(S) / tau + 0.1e1)
/ np.exp(S) * tau ],
[ lamda * tau / mu * (v_x * mu / tau ** 2 - 0.3e1 * np.exp(S)
/ tau ** 3 - 0.1e1) / spd, lamda * tau / mu, 0, -tau / mu * ((v_x
* mu / tau ** 2 - 0.3e1 * np.exp(S) / tau ** 3 - 0.1e1) / spd
+ spd), 0.1e1 / mu],
[ lamda * (-v * tau / kappa * (v_x * mu / tau ** 2 - 0.3e1 * np.exp(S)
/ tau ** 3 - 0.1e1) + tau / kappa * (-spd * (-np.exp(S) / tau ** 2
+ tau - (np.exp(S) / tau + 0.1e1) / tau) + v_x * mu * v / tau ** 2
+ T_x * kappa / tau ** 2 + v * (-0.3e1 * np.exp(S) / tau ** 3
- 0.1e1))) / spd, -lamda * v * tau / kappa, lamda * tau / kappa,
v * tau / kappa * ((v_x * mu / tau ** 2 - 0.3e1 * np.exp(S)
/ tau ** 3 - 0.1e1) / spd + spd) - tau / kappa * ((-spd
* (-np.exp(S) / tau ** 2 + tau - (np.exp(S) / tau + 0.1e1) / tau)
+ v_x * mu * v / tau ** 2 + T_x * kappa / tau ** 2 + v * (-0.3e1
* np.exp(S) / tau ** 3 - 0.1e1)) / spd + spd * v + v_x * mu / tau
- np.exp(S) / tau ** 2 + tau), -v / kappa - tau / kappa * (spd
* (np.exp(S) / tau + 0.1e1) / np.exp(S) * tau - v / tau) ]
])
return out
def A_k(x,lam,s,p):
y = soln([x],sol)
y = y[0]
Ux = profile_ode(x,y,sol,p)
f = 2*y[0]-1-p['Gamma']*p['e_minus']
g = (p['Gamma']*y[1]-(p['nu']+1)*Ux[0])/p['nu']
a21 = lam*y[0]/p['nu']
a22 = y[0]/p['nu']
a23 = y[0]*Ux[0]/p['nu'] - f*Ux[0] - p['Gamma']*Ux[1] # Is the second half correct for U_x_x
a24 = lam*g
a25 = g+Ux[1]/y[0];
a53 = lam*y[0]+p['Gamma']*Ux[0];
a55 = f-lam;
out = np.array([[a22, a23, a24, a25, 0, 0, 0, 0, 0, 0],
[0, 0, lam, 1, 1, 0, 0, 0, 0, 0],
[0, 0, 0, 1, 0, 1, 0, 0, 0, 0],
[p['Gamma'], a53, lam*y[0], a55, 0, 0, 1, 0, 0, 0],
[0, a21, 0, 0, a22, lam, 1, -a24, -a25, 0],
[0, 0, a21, 0, 0, a22, 1, a23, 0, -a25],
[0, 0, 0, a21, a53, lam*y[0], a22 + a55, 0, a23, a24],
[0, 0, 0, 0, 0, 0, 0, 0, 1, -1],
[0, 0, 0, 0, -p['Gamma'], 0, 0, lam*y[0], a55, lam],
[0, 0, 0, 0, 0, -p['Gamma'], 0, -a53, 0, a55]])
return out
def A_k(x, lam, s, p):
y = soln([x], sol)
y = y[0]
Ux = profile_ode(x, y, sol, p)
f = 2 * y[0] - 1 - p['Gamma'] * p['e_minus']
g = (p['Gamma'] * y[1] - (p['nu'] + 1) * Ux[0]) / p['nu']
a21 = lam * y[0] / p['nu']
a22 = y[0] / p['nu']
a23 = y[0] * Ux[0] / p['nu'] - f * Ux[0] - p['Gamma'] * Ux[
1] # Is the second half correct for U_x_x
a24 = lam * g
a25 = g + Ux[1] / y[0]
a53 = lam * y[0] + p['Gamma'] * Ux[0]
a55 = f - lam
out = np.array([[a22, a23, a24, a25, 0, 0, 0, 0, 0, 0],
[0, 0, lam, 1, 1, 0, 0, 0, 0, 0],
[0, 0, 0, 1, 0, 1, 0, 0, 0, 0],
[p['Gamma'], a53, lam * y[0], a55, 0, 0, 1, 0, 0, 0],
[0, a21, 0, 0, a22, lam, 1, -a24, -a25, 0],
[0, 0, a21, 0, 0, a22, 1, a23, 0, -a25],
[0, 0, 0, a21, a53, lam * y[0], a22 + a55, 0, a23, a24],
[0, 0, 0, 0, 0, 0, 0, 0, 1, -1],
[0, 0, 0, 0, -p['Gamma'], 0, 0, lam * y[0], a55, lam],
[0, 0, 0, 0, 0, -p['Gamma'], 0, -a53, 0, a55]])
return out
def A_stable_model_balflux(x,lamda,s,p):
# Balanced flux Evans matrix for the system named stable_model.
#print("entering A_stable_model_balflux")
# wave speed
spd = p['spd']
# parameters
v_neg = p['v_neg']
tau_neg = p['tau_neg']
S_neg = p['S_neg']
m1 = p['m1']
none = p['none']
mu = p['mu']
kappa = p['kappa']
if s['solve'] == 'bvp':
# interpolate profile solution
temp = stablab.soln(x,s)
else:
temp = stablab.deval(x,s.sol)
#profile values
tau = temp[0]
S = temp[1]
ynot = np.array([tau,S])
temp2 = F_stable_model(x,ynot,s,p)
tau_x = temp2[0]
S_x = temp2[1]
v = v_neg - spd * (tau - tau_neg)
v_x = -spd * tau_x
T_x = S_x * np.exp(S) / tau - np.exp(S) / tau ** 2 * tau_x
# Evans matrix -- 5 x 5
out = np.array([
[lamda / spd, 0, 0, -0.1e1 / spd, 0],
[0, 0, 0, 1, 0],
[lamda * (-0.2e1 * np.exp(S) / tau ** 2 + tau) / spd, 0, 0, -(-0.2e1 *
np.exp(S) / tau ** 2 + tau) / spd + v, 1 ],
[lamda * tau / mu * (v_x * mu / tau ** 2 - 0.3e1 * np.exp(S) / tau **
3 - 0.1e1) / spd, lamda * tau / mu, 0, -tau / mu * ((v_x * mu /
tau ** 2 - 0.3e1 * np.exp(S) / tau ** 3 - 0.1e1) / spd + spd),
0.1e1 / mu],
[lamda * (-v * tau / kappa * (v_x * mu / tau ** 2 - 0.3e1 *
np.exp(S) / tau ** 3 - 0.1e1) + tau / kappa * (-spd * (-0.2e1 *
np.exp(S) / tau ** 2 + tau) + v_x * mu * v / tau ** 2 + T_x *
kappa / tau ** 2 + v * (-0.3e1 * np.exp(S) / tau ** 3 - 0.1e1))) /
spd, -lamda * v * tau / kappa, lamda * tau / kappa, v * tau /
kappa * ((v_x * mu / tau ** 2 - 0.3e1 * np.exp(S) / tau ** 3 -
0.1e1) / spd + spd) - tau / kappa * ((-spd * (-0.2e1 * np.exp(S) /
tau ** 2 + tau) + v_x * mu * v / tau ** 2 + T_x * kappa /
tau ** 2 + v * (-0.3e1 * np.exp(S) / tau ** 3 - 0.1e1)) / spd +
spd * v + v_x * mu / tau - np.exp(S) / tau ** 2 + tau), -v /
kappa - tau / kappa * (spd - v / tau)]
])
return out
def A(x, lamda, s, p):
v = soln(x, s)
b = 1 / v[0]
h = 1 - p['a'] * p['gamma'] * v[0]**(-p['gamma'] -
1) + v[1] / (v[0]**2) - lamda / v[0]
out = np.array(
[[0, lamda, 1, 0], [0, 0, 1, 0], [0, 0, 0, 1],
[lamda / p['d'], lamda / p['d'], h / p['d'], (-b / p['d'])]])
return out
def A(x, lamda, s, p):
gamma = p['gamma']
a = p['a']
v = stablab.soln(x, s)
v_prime = profile_ode(x, v, s, p)
v_prime_prime = (1 - a * gamma * v**(-gamma - 1)) * v_prime
A31 = (lamda * a * gamma * v**(-gamma) + a * gamma *
(gamma + 1) * v**(-gamma - 1) * v_prime + v_prime_prime / v -
lamda * v_prime / v - 2 * (v_prime**2) / (v**2))
out = v * np.array([
[-lamda, 0, 1], [0, 0, 1],
[A31, lamda * v, v - a * gamma * v**(-gamma) + 2 * v_prime / v]
]) #,dtype=np.complex)
return out
def error_check_local_model(p,s):
"""
This function error checks our model, looking for values which do not make
physical sense.
"""
x = np.linspace(s.L,s.R,200)
y = np.zeros((len(x),2))
for j in range(len(x)):
temp = soln(x[j],s)
y[j,0] = temp[0]
y[j,1] = temp[1]
# wave speed
spd = p.spd
# values at - infinity
v_neg = p.v_neg
tau_neg = p.tau_neg
S_neg = p.S_neg
# parameters
m1 = p.m1
none = p.none
mu = p.mu
kappa = p.kappa
r = np.zeros((len(x),1))
for j in range(len(x)):
tau = y[j,0]
S = y[j,1]
v = v_neg - spd * (tau - tau_neg)
r[j,0] = v
energy = np.exp(S) / tau + tau ** 2 / 0.2e1
if energy < 0:
raise ValueError('unphysical, energy should be non-negative')
pressure = np.exp(S) / tau ** 2 - tau
if pressure < 0:
raise ValueError('unphysical, pressure should be non-negative')
temperature = np.exp(S) / tau
if temperature < 0:
raise ValueError('unphysical, temperature should be non-negative')
def plot_profile_stable_model(p,s):
"""
Plots the profile equations for the stable model.
"""
x = np.linspace(s.L,s.R,200)
y = np.zeros((len(x),2))
for j in range(len(x)):
temp = stablab.soln(x[j],s)
y[j,0] = temp[0]
y[j,1] = temp[1]
# wave speed
spd = p.spd
# values at - infinity
v_neg = p.v_neg
tau_neg = p.tau_neg
S_neg = p.S_neg
# parameters
m1 = p.m1
none = p.none
mu = p.mu
kappa = p.kappa
r = np.zeros((len(x),1))
w = np.zeros((len(x),1))
for j in range(len(x)):
tau = y[j,0]
S = y[j,1]
v = v_neg - spd * (tau - tau_neg)
r[j,0] = v
T = np.exp(S) / tau
w[j,0] = T
fig = plt.figure("Profile Equations")
plt.plot(x,r)
plt.plot(x,y)
plt.plot(x,w)
plt.xlabel('x')
plt.ylabel('profiles')
plt.legend(['v','tau','S','T'])
plt.show()
def error_check_local_model(p, s):
"""
This function error checks our model, looking for values which do not make
physical sense.
"""
x = np.linspace(s.L, s.R, 200)
y = np.zeros((len(x), 2))
for j in range(len(x)):
temp = soln(x[j], s)
y[j, 0] = temp[0]
y[j, 1] = temp[1]
# wave speed
spd = p.spd
# values at - infinity
v_neg = p.v_neg
tau_neg = p.tau_neg
S_neg = p.S_neg
# parameters
m1 = p.m1
none = p.none
mu = p.mu
kappa = p.kappa
r = np.zeros((len(x), 1))
for j in range(len(x)):
tau = y[j, 0]
S = y[j, 1]
v = v_neg - spd * (tau - tau_neg)
r[j, 0] = v
energy = np.exp(S) / tau + tau**2 / 0.2e1
if energy < 0:
raise ValueError('unphysical, energy should be non-negative')
pressure = np.exp(S) / tau**2 - tau
if pressure < 0:
raise ValueError('unphysical, pressure should be non-negative')
temperature = np.exp(S) / tau
if temperature < 0:
raise ValueError('unphysical, temperature should be non-negative')
def A(x, lam, sol, p):
y = soln([x], sol)
y = y[0]
Ux = profile_ode(x, y, sol, p)
f = 2 * y[0] - 1 - p['Gamma'] * p['e_minus']
g = (p['Gamma'] * y[1] - (p['nu'] + 1) * Ux[0]) / p['nu']
a21 = lam * y[0] / p['nu']
a22 = y[0] / p['nu']
a23 = y[0] * Ux[0] / p['nu'] - f * Ux[0] - p['Gamma'] * Ux[1]
a24 = lam * g
a25 = g + Ux[1] / y[0]
a53 = lam * y[0] + p['Gamma'] * Ux[0]
a55 = f - lam
out = np.array([[0, 1, 0, 0, 0], [a21, a22, a23, a24, a25],
[0, 0, 0, lam, 1], [0, 0, 0, 0, 1],
[0, p['Gamma'], a53, lam * y[0], a55]])
return out
def plot_profile_local_model(p, s):
x = np.linspace(s.L, s.R, 200)
y = np.zeros((len(x), 2))
for j, xVal in enumerate(x):
temp = soln(xVal, s)
y[j, 0] = temp[0]
y[j, 1] = temp[1]
# wave speed
spd = p.spd
# values at - infinity
v_neg = p.v_neg
tau_neg = p.tau_neg
S_neg = p.S_neg
# parameters
m1 = p.m1
none = p.none
mu = p.mu
kappa = p.kappa
r = np.zeros(len(x))
w = np.zeros(len(x))
for j in range(len(x)):
tau = y[j, 0]
S = y[j, 1]
v = v_neg - spd * (tau - tau_neg)
r[j] = v
T = np.exp(S) / tau + 0.1e1
w[j] = T
plt.plot(x, r)
plt.plot(x, y)
plt.plot(x, w)
plt.xlabel('x')
plt.ylabel('profiles')
plt.legend(['v', 'tau', 'S', 'T', 'Location', 'Best'])
plt.show()
def plot_profile_local_model(p,s):
x = np.linspace(s.L,s.R,200)
y = np.zeros((len(x),2))
for j,xVal in enumerate(x):
temp = soln(xVal,s)
y[j,0] = temp[0]
y[j,1] = temp[1]
# wave speed
spd = p.spd
# values at - infinity
v_neg = p.v_neg
tau_neg = p.tau_neg
S_neg = p.S_neg
# parameters
m1 = p.m1
none = p.none
mu = p.mu
kappa = p.kappa
r = np.zeros(len(x))
w = np.zeros(len(x))
for j in range(len(x)):
tau = y[j,0]
S = y[j,1]
v = v_neg - spd * (tau - tau_neg)
r[j] = v
T = np.exp(S) / tau + 0.1e1
w[j] = T
plt.plot(x,r)
plt.plot(x,y)
plt.plot(x,w)
plt.xlabel('x')
plt.ylabel('profiles')
plt.legend(['v','tau','S','T','Location','Best'])
plt.show()
def A(x,lam,sol,p):
y = soln([x],sol)
y = y[0]
Ux = profile_ode(x,y,sol,p)
f = 2*y[0]-1-p['Gamma']*p['e_minus']
g = (p['Gamma']*y[1]-(p['nu']+1)*Ux[0])/p['nu']
a21 = lam*y[0]/p['nu']
a22 = y[0]/p['nu']
a23 = y[0]*Ux[0]/p['nu'] - f*Ux[0] - p['Gamma']*Ux[1]
a24 = lam*g
a25 = g+Ux[1]/y[0];
a53 = lam*y[0]+p['Gamma']*Ux[0];
a55 = f-lam;
out = np.array([[0, 1, 0, 0, 0],
[a21, a22, a23, a24, a25],
[0, 0, 0, lam, 1],
[0, 0, 0, 0, 1],
[0, p['Gamma'], a53, lam*y[0], a55]])
return out
'Flinear': profile_jacobian, # J is the profile ode Jacobian
'UL': np.array([p['v_minus'],p['e_minus']]), # These are the endstates of the profile and its derivative at x = -infty
'UR': np.array([p['v_plus'],p['e_plus']]), # These are the endstates of the profile and its derivative at x = +infty
'tol': 1e-7
})
sol.update({
'phase': 0.5*(sol['UL']+sol['UR']), # this is the phase condition for the profile at x = 0
'order': [1], # this indicates to which component the phase conditions is applied
'stats': 'on', # this prints data and plots the profile as it is solved
'bvp_options': {'Tol': 1e-6, 'Nmax': 200}
})
p,s = profile_flux(p,sol)
x = np.linspace(s['L'],s['R'],200)
y = soln(x,s)
plt.figure("Profile")
plt.plot(x,y)
plt.show()
# set STABLAB structures to local default values
#s, e, m, c = emcset(s, 'front', [2,3], 'reg_adj_compound', A, A_k)
s, e, m, c = emcset(s,'front',[2,3],'reg_reg_polar',A)
circpnts, imagpnts, innerpnts = 20, 30, 5
r = 10
spread = 4
zerodist = 10**(-2)
# ksteps, lambda_steps = 32, 0
preimage = semicirc2(circpnts, imagpnts, innerpnts, c['ksteps'],
def A_local_model_balflux(x, lamda, s, p):
"""
Balanced flux Evans matrix for the system named local_model.
"""
# wave speed
spd = p['spd']
# parameters
v_neg = p['v_neg']
tau_neg = p['tau_neg']
S_neg = p['S_neg']
m1 = p['m1']
none = p['none']
mu = p['mu']
kappa = p['kappa']
# interpolate profile solution
temp = soln(x, s)
#profile values
tau = temp[0]
S = temp[1]
ynot = np.array([tau, S])
temp2 = F_local_model(x, ynot, s, p)
tau_x = temp2[0]
S_x = temp2[1]
v = v_neg - spd * (tau - tau_neg)
v_x = -spd * tau_x
T_x = S_x * np.exp(S) / tau - np.exp(S) / tau**2 * tau_x
# Evans matrix
out = np.array(
[[lamda / spd, 0, 0, -0.1e1 / spd, 0], [0, 0, 0, 1, 0],
[
lamda * (-np.exp(S) / tau**2 + tau -
(np.exp(S) / tau + 0.1e1) / tau) / spd, 0, 0,
-(-np.exp(S) / tau**2 + tau -
(np.exp(S) / tau + 0.1e1) / tau) / spd + v,
(np.exp(S) / tau + 0.1e1) / np.exp(S) * tau
],
[
lamda * tau / mu *
(v_x * mu / tau**2 - 0.3e1 * np.exp(S) / tau**3 - 0.1e1) / spd,
lamda * tau / mu, 0, -tau / mu *
((v_x * mu / tau**2 - 0.3e1 * np.exp(S) / tau**3 - 0.1e1) / spd +
spd), 0.1e1 / mu
],
[
lamda *
(-v * tau / kappa *
(v_x * mu / tau**2 - 0.3e1 * np.exp(S) / tau**3 - 0.1e1) +
tau / kappa * (-spd * (-np.exp(S) / tau**2 + tau -
(np.exp(S) / tau + 0.1e1) / tau) +
v_x * mu * v / tau**2 + T_x * kappa / tau**2 + v *
(-0.3e1 * np.exp(S) / tau**3 - 0.1e1))) / spd,
-lamda * v * tau / kappa, lamda * tau / kappa, v * tau / kappa *
((v_x * mu / tau**2 - 0.3e1 * np.exp(S) / tau**3 - 0.1e1) / spd +
spd) - tau / kappa *
((-spd * (-np.exp(S) / tau**2 + tau -
(np.exp(S) / tau + 0.1e1) / tau) +
v_x * mu * v / tau**2 + T_x * kappa / tau**2 + v *
(-0.3e1 * np.exp(S) / tau**3 - 0.1e1)) / spd + spd * v +
v_x * mu / tau - np.exp(S) / tau**2 + tau),
-v / kappa - tau / kappa *
(spd * (np.exp(S) / tau + 0.1e1) / np.exp(S) * tau - v / tau)
]])
return out
s.UL = np.array([1
]) # These are the endstates of the profile at x = -infty
s.UR = np.array([p.vp
]) # These are the endstates of the profile at x = +infty
s.phase = 0.5 * (s.UL + s.UR
) # this is the phase condition for the profile at x = 0
s.order = [
0
] # this indicates to which componenet the phase conditions is applied
s.stats = 'on' # this prints data and plots the profile as it is solved
s.bvp_options = {'Tol': 1e-9}
p, s = stablab.profile_flux(p, s) # solve profile for first time
# plot the profile
x = np.linspace(s.L, s.R, 200)
y = stablab.soln(x, s)
plt.plot(x, y)
plt.show()
# This choice solves the right hand side via exterior products
s.A = A
s.Ak = Ak
#s,e,m,c = stablab.emcset(s, 'front', [1,2], 'reg_adj_compound', A, Ak)
s, e, m, c = stablab.emcset(s, 'front', [1, 2], 'reg_reg_polar', A)
#s,e,m,c = emcset(s,'front',[1,2],'reg_adj_polar')
m.options = {}
m.options['AbsTol'] = 1e-9
m.options['RelTol'] = 1e-9
# display a waitbar
c.stats = 'print' # 'on', 'print', or 'off'
0
] # this indicates to which componenet the phase conditions is applied
s.stats = 'on' # this prints data and plots the profile as it is solved
if j == 0:
# there are some other options you specify. You can look in profile_flux to see them
p, s = profile_flux(p, s) # solve profile for first time
s_old = s
else:
# this time we are using continuation
p, s = profile_flux(p, s, s_old)
# solve profile
# plot the profile
x = np.linspace(s.L, s.R, 200)
y = soln(x, s)
plt.title("Profile")
plt.plot(x, y[:, 0])
plt.plot(x, y[:, 1])
plt.show()
# structure variables
# Here you can choose the method you use for the Evans function, or you can set the option
# to default and let it choose. [2,2] is the size of the manifold evolving from - and + infy in the
# Evans solver. 'front' indicates you are solving for a traveling wave front and not a periodic solution
# s,e,m,c = emcset(s,'front',LdimRdim(A,s,p),'default') # default for capillarity is reg_reg_polar
# s,e,m,c = emcset(s,'front',[2,2],'reg_adj_polar')
# s,e,m,c = emcset(s,'front',[2,2],'adj_reg_polar')
# s,e,m,c = emcset(s,'front',[2,2],'reg_reg_polar')
