def combustion_profile_continuation():
s = {'I':16,'R':16,'L':-32,'side':1,'n':4}
p = {'beta': 3.5, 'tau': .1}
s['L'], s['R'] = -8.0*np.exp(0.54*p['beta']), 8.0*np.exp(0.54*p['beta'])
beta_vals = [3.5, 4, 5, 6, 7, 8, 8.5, 9, 9.5, 10, 10.5, 11, 11.5, 12, 12.5, 13, 13.5] +list(np.linspace(14,18,40))
options = struct()
# options include abstol, reltol, singularterm, stats, vectorized, maxnewpts,slopeout,xint
options.fjacobian = lambda x,y: double_F_jacobian(x,y,s,p)
options.bcjacobian = lambda x,y: bc2_jacobian(x,y,s,p)
options.abstol = 1e-7
options.reltol = 1e-6
options.nmax = 5000
options.stats='on'
solinit = bvpinit(np.linspace(0,s['I'],100),lambda x: init_prof_guess(x,s,p))
s['sol'] = bvp6c(lambda x,y: double_profile_F(x,y,s,p),
lambda ya,yb: profile_bcs(ya,yb,s,p),
solinit,options)
xint = np.linspace(0,s['I'],100); Sxint,_ = deval(s['sol'],xint)
for j in range(len(beta_vals)):
p['beta']=beta_vals[j]
s['L']=-8.0*np.exp(0.54*p['beta'])
s['R']=8.0*np.exp(0.54*p['beta'])
options.fjacobian = lambda x,y: double_F_jacobian(x,y,s,p)
options.bcjacobian = lambda x,y: bc2_jacobian(x,y,s,p)
solinit = bvpinit(xint,Sxint)
s['sol'] = bvp6c(lambda x,y: double_profile_F(x,y,s,p),
lambda ya,yb: profile_bcs(ya,yb,s,p),
solinit,options)
print "p['beta'] = ", p['beta']
Sxint,_ = deval(s['sol'],xint)
plot_soln(xint,Sxint,s,p,save=True)
def combustion():
s = {'I':16,'R':16,'L':-32,'side':1,'n':4}
p = {'beta':3.5, 'tau': .1}
options = struct()
# options include abstol, reltol, singularterm, stats, vectorized, maxnewpts,slopeout,xint
options.fjacobian = lambda x,y: double_F_jacobian(x,y,s,p)
options.bcjacobian = lambda x,y: bc2_jacobian(x,y,s,p)
options.abstol = 1e-7
options.reltol = 1e-6
options.nmax = 5000
options.stats='on'
solinit = bvpinit(np.linspace(0,s['I'],100),lambda x: init_prof_guess(x,s,p))
s['sol'] = bvp6c(lambda x,y: double_profile_F(x,y,s,p),
lambda ya,yb: profile_bcs(ya,yb,s,p),
solinit,options)
xint = np.linspace(0,s['I'],400); Sxint,_ = deval(s['sol'],xint)
plot_soln(xint,Sxint,s,p)
def combustion_roottracking():
#
# Parameters
#
p={'beta':3.5,'tau': 0.1,'temp_n':200,'rooot':1e-4*(.12375+ 6.3313j)} #'rooot' for beta = 7.1
#
# Structure variables
#
s={'I':16.0,'R':16.0,'L':-32.0, 'side':1,'n':4, 'rarray':range(0,4),
'larray':range(4,8)}
s['Grarray'] = range(0,8)
s['Glarray'] = range(8,16)
s['L'], s['R']=-8.0*np.exp(0.54*p['beta']), 8.0*np.exp(0.54*p['beta'])
s['PFunctions'] = 'combustionEvans' # Must be called before calling emcset
[s,e,m,c] = emcset(s,'front',2,2,'reg_reg_polar')
m['method'] = mod_drury; m['options'] = {'RelTol':1e-8,'AbsTol':1e-8}
''' Profile computation; Continuation of traveling wave. '''
options_F = struct()
# options include abstol, reltol, singularterm, stats, vectorized, maxnewpts,slopeout,xint
options_F.abstol, options_F.reltol = 1e-10, 1e-9
options_F.fjacobian = lambda x,y: double_F_jacobian(x,y,s,p)
options_F.bcjacobian = lambda x,y: bc2_jacobian(x,y,s,p)
options_F.nmax = 5000
options_F.stats='on'
xint = np.linspace(0,s['I'],100)
solinit = bvpinit(xint,lambda x: init_prof_guess(x,s,p))
s['solution'] = bvp6c(lambda x,y: double_profile_F(x,y,s,p),
lambda ya,yb: profile_bcs(ya,yb,s,p),
solinit,options_F)
Sxint,_ = deval(s['solution'],xint)
plot_soln(xint,Sxint,s,p)
beta_vals = [3.5, 4, 5, 6, 7, 7.1]
for j in range(len(beta_vals)):
p['beta']=beta_vals[j]
s['L']=-8.0*np.exp(0.54*p['beta'])
s['R']=8.0*np.exp(0.54*p['beta'])
options_F.fjacobian = lambda x,y: double_F_jacobian(x,y,s,p)
options_F.bcjacobian = lambda x,y: bc2_jacobian(x,y,s,p)
solinit = bvpinit(xint,Sxint)
s['solution'] = bvp6c(lambda x,y: double_profile_F(x,y,s,p),
lambda ya,yb: profile_bcs(ya,yb,s,p),
solinit,options_F)
print "beta = ", p['beta']
Sxint,_ = deval(s['solution'],xint)
plot_soln(xint,Sxint,s,p,save=True)
s['eigf']=True
s, x1, x2 = EvansDF.eigf_init_guess(s,e,m,c,p)
##############################################################################
if s['eigf']:
for j in range(0,m['n']):
plt.plot(x1,np.flipud(np.real(s['guess'][j+m['n'],:])) ,'k')
plt.plot(np.flipud(x2), np.real(s['guess'][j,:]),'k')
plt.title('Eigenfunction for the combustion equation')
plt.xlabel('x'); plt.ylabel('W (Initial Guess: Hello)');
format = '.eps'
plt.savefig(package_directory+'/combustion/data_combustion/'+
'combustion_eigenfunction_guess'+format)
# plt.show()
n=s['n']; eig_n = 2*n
s['guess'] = np.concatenate( ( np.real(s['guess'][0:n,:]),
np.imag(s['guess'][0:n,:]),
np.real(s['guess'][n:eig_n,:]),
np.imag(s['guess'][n:eig_n,:]),
np.array([np.real(s['guess'][eig_n,:]) ]),
np.array([np.imag(s['guess'][eig_n,:]) ]) ),
axis=0 )
s['ph'] = np.array([[0,s['guess'][0,0] ],
[4,s['guess'][4,0] ] ])
del n,eig_n
options_eigfF = struct()
# options include abstol, reltol, singularterm, stats, vectorized, maxnewpts,slopeout,xint
options_eigfF.abstol, options_eigfF.reltol = 1e-10, 1e-9
options_eigfF.fjacobian = lambda x,y: double_F_jacobian(x,y,s,p)
options_eigfF.bcjacobian = lambda x,y: bc2_jacobian(x,y,s,p)
options_eigfF.nmax = 5000
options_eigfF.stats='on'
data = []
beta_vals = list(np.linspace(7.1,18,219))
total_time = 0
import time
start = time.time()
for j in range(len(beta_vals)):
p['beta']=beta_vals[j]
s['L']=-8.0*np.exp(0.54*p['beta'])
s['R']=8.0*np.exp(0.54*p['beta'])
options_F.fjacobian = lambda x,y: double_F_jacobian(x,y,s,p)
options_F.bcjacobian = lambda x,y: bc2_jacobian(x,y,s,p)
solinit = bvpinit(xint,Sxint)
s['solution'] = bvp6c(lambda x,y: double_profile_F(x,y,s,p),
lambda ya,yb: profile_bcs(ya,yb,s,p),
solinit,options_F)
Sxint,_ = deval(s['solution'],xint)
options_eigfF.fjacobian = lambda x,y: double_eigfF_jacobian(x,y,s,p)
options_eigfF.bcjacobian = lambda x,y: eigfF_bcs_jacobian(x,y,s,p)
if j ==0:
solinit = bvpinit(np.linspace(0,s['I'],len(s['guess'][1])),s['guess'])
else:
solinit = bvpinit(xint,eigf_Sxint)
s['eigf_solution'] = bvp6c(lambda x,y: eigf_F(x,y,s,p),
lambda ya,yb: eigf_bcs(ya,yb,s,p),
solinit,options_eigfF)
lmbda = s['eigf_solution'].y[16,-1] + 1.j*s['eigf_solution'].y[17,-1]
end = time.time()
total_time += end-start
start = end
# print "beta = ", p['beta']
# print "lambda = ", lmbda
data.append((p['beta'],lmbda))
print "total_time = ", total_time
print data
eigf_Sxint,_ = deval(s['eigf_solution'],xint)
plot_soln(xint,Sxint,s,p,save=True)
plot_soln_eigf(xint,eigf_Sxint,s,p,save=True)
return s,p
def test_ndtools_prob31(flag=False):
# prob31 using numdifftools to construct a Jacobian
import numdifftools as nd
cos, sin, tan = math.cos, math.sin, math.tan
sec = lambda x: 1./cos(x)
# Parameters
epsilon = .05
alpha, beta = 0., 0.
# ------------------------------------------------------------
def ode(x,y):
return np.array([ sin(y[1]),
y[2],
-epsilon**(-1)*y[3],
epsilon**(-1)*( (y[0]-1)*cos(y[1])- y[2]*(sec(y[1]) + epsilon*y[3]*tan(y[1])) )
]);
# ------------------------------------------------------------
# def f_jacobian(x,y):
# return np.array([ [0, cos(y[1]), 0, 0],
# [0, 0, 1., 0],
# [0, 0, 0, -epsilon**(-1.)],
# [epsilon**(-1.)*cos(y[1]),
# epsilon**(-1)*( (1-y[0])*sin(y[1])- y[2]*(sec(y[1])*tan(y[1]) + epsilon*y[3]*(sec(y[1]))**2.) ),
# -epsilon**(-1)*(sec(y[1]) + epsilon*y[3]*tan(y[1])),
# -y[2]*tan(y[1])]
# ])
def f_jacobian(x,y):
g = nd.Jacobian(lambda z: ode(x,z) )
return g(y)
# ------------------------------------------------------------
def bc_jacobian(x,y):
dGdya = np.array([ [1, 0, 0, 0],
[0, 0, 1, 0],
[0, 0, 0, 0],
[0, 0, 0, 0] ]);
dGdyb = np.array([ [0, 0, 0, 0],
[0, 0, 0, 0],
[1, 0, 0, 0],
[0, 0, 1, 0] ]);
return dGdya,dGdyb
# ------------------------------------------------------------
def bcs(ya,yb):
return np.array([ ya[0] - alpha,
ya[2] - alpha,
yb[0] - beta,
yb[2] - beta
]);
# ------------------------------------------------------------
def init(x):
return np.array([ -(1/5.)*(x-.5)**2.+.05,
(16/5.)*(x-.5)**3./3.-.8*x+.45,
(16/5.)*(x-.5)**2.-.8,
-epsilon*(32/5.)*(x-.5)
]);
# ------------------------------------------------------------
options = struct()
# options include abstol, reltol, singularterm, stats, vectorized, maxnewpts,slopeout,xint
options.abstol, options.reltol = 1e-8, 1e-7
options.fjacobian, options.bcjacobian = f_jacobian, bc_jacobian
options.nmax = 20000
solinit = bvpinit(np.linspace(0,1,10),init)
with nostdout():
sol = bvp6c(ode,bcs,solinit,options)
xint = np.linspace(0,1,100); Sxint,_ = deval(sol,xint)
# savemat('np_vector.mat', {'x':sol.x,'y':sol.y,'xint':xint,'Sxint':Sxint})
m_data = loadmat(port_path+'test_data/bvp_prob31.mat')
grid_diff = np.max(np.abs(sol.x-m_data['x']))
sol_diff = np.max(np.abs(sol.y-m_data['y']))
grid_eval = np.max(np.abs(xint-m_data['xint']))
sol_eval = np.max(np.abs(Sxint-m_data['Sxint']))
if flag==True:
for j in range(4):
plt.plot(sol.x,sol.y[j],linewidth=2.0)
# plt.axis([-0.02, 1.02, .95, 1.8])
plt.title('Numerical Solution')
plt.xlabel('x'); plt.ylabel('y'); plt.show()
assert grid_diff < 1e-10,"Difference in grids = %e"%grid_diff
assert sol_diff < 1e-10, "Difference in solutions = %e"%sol_diff
assert grid_eval < 1e-10, "Difference in xint = %e"%grid_eval
assert sol_eval < 1e-10, "Difference in Sxint = %e"%sol_eval
print("Difference in grids = %e"%grid_diff)
print("Difference in solutions = %e"%sol_diff)
print("Difference in xint = %e"%grid_eval)
print("Difference in Sxint = %e"%sol_eval)
print("bvp_prob31 passes\n\n\n")
return (grid_diff, sol_diff,grid_eval, sol_eval)
def measles_seir(flag=False):
# prob31 using numdifftools to construct a Jacobian
# cos, sin, tan = math.cos, math.sin, math.tan
# sec = lambda x: 1./cos(x)
beta0 = 1575
beta1 = 1.
eta = 0.01
lmbda = .0279
mu = .02
# print(mu)
def beta(t):
return beta0*(1. + beta1*cos(2*pi*t))
# ------------------------------------------------------------
def ode(x,y):
return np.array([ mu - beta(x)*y[0]*y[2],
beta(x)*y[0]*y[2] - y[1]/lmbda,
y[1]/lmbda - y[2]/eta,
0.,
0.,
0.
]);
# ------------------------------------------------------------
def f_jacobian(x,y):
out = np.array([ [ -beta(x)*y[2], 0, -beta(x)*y[0], 0,0,0 ],
[ beta(x)*y[2], -1./lmbda, beta(x)*y[0], 0,0,0 ],
[ 0., 1./lmbda, -1/eta, 0,0,0 ],
[0.,0.,0.,0.,0.,0.,],
[0.,0.,0.,0.,0.,0.,],
[0.,0.,0.,0.,0.,0.,] ])
return out
# ------------------------------------------------------------
def bc_jacobian(x,y):
dGdya = np.array([ [1, 0, 0, -1, 0, 0],
[0, 1, 0, 0, -1, 0],
[0, 0, 1, 0, 0, -1],
[0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0] ]);
dGdyb = np.array([ [0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0],
[1, 0, 0, -1, 0, 0],
[0, 1, 0, 0, -1, 0],
[0, 0, 1, 0, 0, -1] ]);
return dGdya, dGdyb
# ------------------------------------------------------------
def bcs(ya,yb):
return np.array([ ya[0] - ya[3],
ya[1] - ya[4],
ya[2] - ya[5],
yb[0] - yb[3],
yb[1] - yb[4],
yb[2] - yb[5]
]);
# ------------------------------------------------------------
# def init(x):
# return np.array([ -(1/5.)*(x-.5)**2.+.05,
# (16/5.)*(x-.5)**3./3.-.8*x+.45,
# (16/5.)*(x-.5)**2.-.8,
# -epsilon*(32/5.)*(x-.5)
# ]);
def init(x):
S = .1 + .05 * np.cos(2. * pi * x)
out = np.ones((6,len(x)))
out[0] = .65 + .4*S
out[1] = .05 * (1. - S)-.042
out[2] = .05 * (1. - S)-.042
out[3] = .075*out[3]
out[4] = .005*out[4]
out[5] = .005*out[5]
return out
# ------------------------------------------------------------
if flag==False:
x = np.linspace(0,1,100)
guess = init(x)
plt.plot(x,guess[0],'-k')
# plt.plot(x,guess[1],'-k')
# plt.plot(x,guess[2],'-k')
# plt.plot(x,guess[3],'-r')
# plt.plot(x,guess[4],'-r')
# plt.plot(x,guess[5],'-r')
plt.show()
def init(x):
S = .1 + .05 * cos(2. * pi * x)
# out = np.ones((6,len(x)))
# out[0] = .5*S
# out[1] = .05 * (1. - S)
# out[2] = .05 * (1. - S)
# out[3] = .08*out[3]
# out[4] = .08*out[4]
# out[5] = .08*out[5]
return np.array([.65 + .4*S, .05 * (1. - S)-.042, .05 * (1. - S)-.042, .075,.005,.005])
options = struct()
# options include abstol, reltol, singularterm, stats, vectorized, maxnewpts,slopeout,xint
options.abstol, options.reltol = 1e-8, 1e-7
options.fjacobian, options.bcjacobian = f_jacobian, bc_jacobian
options.nmax = 20000
solinit = bvpinit(np.linspace(0,1,100),init)
# with nostdout():
sol = bvp6c(ode,bcs,solinit,options)
xint = np.linspace(0,1,100); Sxint,_ = deval(sol,xint)
if flag==True:
for j in range(3):
plt.plot(sol.x,sol.y[j],linewidth=2.0)
plt.title('Numerical Solution')
plt.xlabel('x'); plt.ylabel('y'); plt.show()
return
def prob24(flag=False):
# Parameters
epsilon, gamma = .008, 1.4
alpha, beta = .9129, .375
# ------------------------------------------------------------
def ode(x,y):
return np.array([ y[1],
1./(1+x**2.)*((1+gamma)/(2*epsilon)-2*x)*y[1]-y[1]*y[0]**(-2.)/(epsilon*(1+x**2.))-
2.*x/(epsilon*(1+x**2.)**2.)*(y[0]**(-1.)-(gamma-1)/2.*y[0])
]);
# ------------------------------------------------------------
def f_jacobian(x,y):
return np.array([ [0, 1],
[ 2.*y[1]*y[0]**(-3.)/(epsilon*(1+x**2))+epsilon**(-1)*2*x*(1+x**2)**(-2)*(y[0]**(-2)+(gamma-1)/2.) ,
(1+x**2.)**(-1)*(epsilon**(-1)*(1+gamma)/2.-2*x)-y[0]**(-2.)/(epsilon*(1+x**2)) ]
])
# ------------------------------------------------------------
def bc_jacobian(x,y):
dGdya = np.array([ [1, 0],
[0, 0] ]);
dGdyb = np.array([ [0, 0],
[1, 0] ]);
return dGdya,dGdyb
# ------------------------------------------------------------
def bcs(ya,yb):
return np.array([ ya[0] - alpha,
yb[0] - beta ]);
# ------------------------------------------------------------
def init(x):
return np.array([ (.375-.9129)*x+.9129,
(.375-.9129) ]);
# ------------------------------------------------------------
options = struct()
# options include abstol, reltol, singularterm, stats, vectorized, maxnewpts,slopeout,xint
options.abstol, options.reltol = 1e-8, 1e-7
options.fjacobian, options.bcjacobian = f_jacobian, bc_jacobian
options.nmax = 20000
solinit = bvpinit(np.linspace(0,1,80),init)
with nostdout():
sol = bvp6c(ode,bcs,solinit,options)
xint = np.linspace(0,1,100); Sxint,_ = deval(sol,xint)
# savemat('np_vector.mat', {'x':sol.x,'y':sol.y,'xint':xint,'Sxint':Sxint})
m_data = loadmat(port_path+'test_data/bvp_prob24.mat')
grid_diff = np.max(np.abs(sol.x-m_data['x']))
sol_diff = np.max(np.abs(sol.y-m_data['y']))
grid_eval = np.max(np.abs(xint-m_data['xint']))
sol_eval = np.max(np.abs(Sxint-m_data['Sxint']))
if flag==True:
# plt.plot(sol.x,sol.y[0],linewidth=2.0)
# # plt.axis([-0.02, 1.02, .95, 1.8])
# plt.title('Numerical Solution')
# plt.xlabel('x'); plt.ylabel('y'); plt.show()
assert grid_diff < 1e-10,"Difference in grids = %e"%grid_diff
assert sol_diff < 1e-10, "Difference in solutions = %e"%sol_diff
assert grid_eval < 1e-10, "Difference in xint = %e"%grid_eval
assert sol_eval < 1e-10, "Difference in Sxint = %e"%sol_eval
print("Difference in grids = %e"%grid_diff)
print("Difference in solutions = %e"%sol_diff)
print("Difference in xint = %e"%grid_eval)
print("Difference in Sxint = %e"%sol_eval)
print("bvp_prob24 passes\n\n\n")
return (grid_diff, sol_diff,grid_eval, sol_eval)
def prob27(flag=False):
# Parameters
epsilon = .005
alpha, beta = 1., 1/3.
# ------------------------------------------------------------
def ode(x,y):
return np.array([ y[1],
1./epsilon*y[0]*(1.-y[1]) ]);
# ------------------------------------------------------------
def f_jacobian(x,y):
return np.array([ [0, 1],
[1./epsilon*(1-y[1]),-y[0]/epsilon] ])
# ------------------------------------------------------------
def bc_jacobian(x,y):
dGdya = np.array([ [1, 0],
[0, 0] ]);
dGdyb = np.array([ [0, 0],
[1, 0] ]);
return dGdya,dGdyb
# ------------------------------------------------------------
def bcs(ya,yb):
return np.array([ ya[0] - alpha,
yb[0] - beta ]);
# ------------------------------------------------------------
def init(x):
return np.array([ 8/3.*x**2.-10/3.*x+1.,
16/3.*x-10/3. ]);
# ------------------------------------------------------------
options = struct()
# options include abstol, reltol, singularterm, stats, vectorized, maxnewpts,slopeout,xint
options.abstol, options.reltol = 1e-8, 1e-7
options.fjacobian, options.bcjacobian = f_jacobian, bc_jacobian
options.nmax = 20000
solinit = bvpinit(np.linspace(0,1,30),init)
with nostdout():
sol = bvp6c(ode,bcs,solinit,options)
xint = np.linspace(0,1,100); Sxint,_ = deval(sol,xint)
# savemat('np_vector.mat', {'x':sol.x,'y':sol.y,'xint':xint,'Sxint':Sxint})
m_data = loadmat(port_path+'test_data/bvp_prob27.mat')
grid_diff = np.max(np.abs(sol.x-m_data['x']))
sol_diff = np.max(np.abs(sol.y-m_data['y']))
grid_eval = np.max(np.abs(xint-m_data['xint']))
sol_eval = np.max(np.abs(Sxint-m_data['Sxint']))
if flag==True:
# plt.plot(sol.x,sol.y[0],linewidth=2.0)
# # plt.axis([-0.02, 1.02, .95, 1.8])
# plt.title('Numerical Solution')
# plt.xlabel('x'); plt.ylabel('y'); plt.show()
assert grid_diff < 1e-10,"Difference in grids = %e"%grid_diff
assert sol_diff < 1e-10, "Difference in solutions = %e"%sol_diff
assert grid_eval < 1e-10, "Difference in xint = %e"%grid_eval
assert sol_eval < 1e-10, "Difference in Sxint = %e"%sol_eval
print("Difference in grids = %e"%grid_diff)
print("Difference in solutions = %e"%sol_diff)
print("Difference in xint = %e"%grid_eval)
print("Difference in Sxint = %e"%sol_eval)
print("bvp_prob27 passes\n\n\n")
return (grid_diff, sol_diff,grid_eval, sol_eval)
def prob1(flag=False):
# BVP parameters
epsilon = .01
# ------------------------------------------------------------
def analytic_sol(x):
out = np.exp(-x/math.sqrt(epsilon)) - np.exp((x-2)/math.sqrt(epsilon));
return out/(1. - np.exp(-2./math.sqrt(epsilon)));
# ------------------------------------------------------------
def ode(x,y):
return np.array([ y[1],
y[0]/epsilon ]);
# ------------------------------------------------------------
def f_jacobian(x,y):
return np.array([ [0, 1],
[1./epsilon,0] ])
# ------------------------------------------------------------
def bc_jacobian(x,y):
dGdya = np.array([ [1, 0],
[0, 0] ]);
dGdyb = np.array([ [0, 0],
[1, 0] ]);
return dGdya,dGdyb
# ------------------------------------------------------------
def bcs(ya,yb):
return np.array([ ya[0]-1,
yb[0] ]);
# ------------------------------------------------------------
def init(x):
return np.array([ 1.-x,
-1. ]);
# ------------------------------------------------------------
options = struct()
# options include abstol, reltol, singularterm, stats, vectorized, maxnewpts,slopeout,xint
options.abstol, options.reltol = 1e-8, 1e-7
options.fjacobian, options.bcjacobian = f_jacobian, bc_jacobian
options.nmax = 20000
solinit = bvpinit(np.linspace(0,1,10),init)
with nostdout():
sol = bvp6c(ode,bcs,solinit,options)
xint = np.linspace(0,1,100); Sxint,_ = deval(sol,xint)
# savemat('np_vector.mat', {'x':sol.x,'y':sol.y,'xint':xint,'Sxint':Sxint})
m_data = loadmat(port_path+'test_data/bvp_prob1.mat')
try:
grid_diff = np.max(np.abs(sol.x-m_data['x']))
sol_diff = np.max(np.abs(sol.y-m_data['y']))
grid_eval = np.max(np.abs(xint-m_data['xint']))
sol_eval = np.max(np.abs(Sxint-m_data['Sxint']))
except:
raise ValueError
if flag==True:
# plt.plot(xint,Sxint[0],linewidth=2.0)
# plt.axis([0, 1, -.1, 1.1])
# plt.title('Numerical Solution')
# plt.xlabel('x'); plt.ylabel('y'); plt.show()
assert grid_diff < 1e-14,"Difference in grids = %e"%grid_diff
assert sol_diff < 1e-14, "Difference in solutions = %e"%sol_diff
assert grid_eval < 1e-14, "Difference in xint = %e"%grid_eval
assert sol_eval < 1e-14, "Difference in Sxint = %e"%sol_eval
print("Difference in grids = %e"%grid_diff)
print("Difference in solutions = %e"%sol_diff)
print("Difference in xint = %e"%grid_eval)
print("Difference in Sxint = %e"%sol_eval)
print("bvp_prob1 passes\n\n\n")
return (grid_diff, sol_diff, grid_eval, sol_eval)
def prob20(flag=False):
# exp, pi = math.exp, math.pi
# cos, sin = math.cos, math.sin
# log, cosh = math.log, math.cosh
# BVP parameters
epsilon = .06 # .5, .3, .06
alpha = 1 + epsilon*math.log(math.cosh(-.745/epsilon))
beta = 1 + epsilon*math.log(math.cosh(.255/epsilon))
# ------------------------------------------------------------
# def analytic_sol(x):
# out = np.exp(-x/math.sqrt(epsilon)) - np.exp((x-2)/math.sqrt(epsilon));
# return out/(1. - np.exp(-2./math.sqrt(epsilon)));
# ------------------------------------------------------------
def ode(x,y):
return np.array([ y[1],
-y[1]**2./epsilon + 1./epsilon ]);
# ------------------------------------------------------------
def f_jacobian(x,y):
return np.array([ [0, 1],
[0.,-2.*y[1]/epsilon] ])
# ------------------------------------------------------------
def bc_jacobian(x,y):
dGdya = np.array([ [1, 0],
[0, 0] ]);
dGdyb = np.array([ [0, 0],
[1, 0] ]);
return dGdya,dGdyb
# ------------------------------------------------------------
def bcs(ya,yb):
return np.array([ ya[0] - alpha,
yb[0] - beta ]);
# ------------------------------------------------------------
def init(x):
return np.array([ 1.-x,
-1. ]);
# ------------------------------------------------------------
options = struct()
# options include abstol, reltol, singularterm, stats, vectorized, maxnewpts,slopeout,xint
options.abstol, options.reltol = 1e-8, 1e-7
options.fjacobian, options.bcjacobian = f_jacobian, bc_jacobian
options.nmax = 20000
solinit = bvpinit(np.linspace(0,1,10),init)
with nostdout():
sol = bvp6c(ode,bcs,solinit,options)
xint = np.linspace(0,1,100); Sxint,_ = deval(sol,xint)
# savemat('np_vector.mat', {'x':sol.x,'y':sol.y,'xint':xint,'Sxint':Sxint})
m_data = loadmat(port_path+'test_data/bvp_prob20.mat')
grid_diff = np.max(np.abs(sol.x-m_data['x']))
sol_diff = np.max(np.abs(sol.y-m_data['y']))
grid_eval = np.max(np.abs(xint-m_data['xint']))
sol_eval = np.max(np.abs(Sxint-m_data['Sxint']))
if flag==True:
# print(sol.x.shape,sol.y[0].shape)
# plt.plot(sol.x,sol.y[0],linewidth=2.0)
# plt.axis([-0.02, 1.02, .95, 1.8])
# plt.title('Numerical Solution')
# plt.xlabel('x'); plt.ylabel('y'); plt.show()
assert grid_diff < 1e-10,"Difference in grids = %e"%grid_diff
assert sol_diff < 1e-10, "Difference in solutions = %e"%sol_diff
assert grid_eval < 1e-10, "Difference in xint = %e"%grid_eval
assert sol_eval < 1e-10, "Difference in Sxint = %e"%sol_eval
print("Difference in grids = %e"%grid_diff)
print("Difference in solutions = %e"%sol_diff)
print("Difference in xint = %e"%grid_eval)
print("Difference in Sxint = %e"%sol_eval)
print("bvp_prob20 passes\n\n\n")
return(grid_diff, sol_diff,grid_eval, sol_eval)
def prob7(flag=False):
exp, pi = math.exp, math.pi
cos, sin = math.cos, math.sin
# BVP parameters
epsilon = .0001 # .001, .0001
# ------------------------------------------------------------
# def analytic_sol(x):
# out = np.exp(-x/math.sqrt(epsilon)) - np.exp((x-2)/math.sqrt(epsilon));
# return out/(1. - np.exp(-2./math.sqrt(epsilon)));
# ------------------------------------------------------------
def ode(x,y):
f = -epsilon**(-1.)*( (1. + epsilon*pi**2.)*cos(pi*x) + pi*x*sin(pi*x) )
return np.array([ y[1],
-x*y[1]/epsilon + 1./epsilon*y[0] + f ]);
# ------------------------------------------------------------
def f_jacobian(x,y):
return np.array([ [0, 1],
[1./epsilon,-x/epsilon] ])
# ------------------------------------------------------------
def bc_jacobian(x,y):
dGdya = np.array([ [1, 0],
[0, 0] ]);
dGdyb = np.array([ [0, 0],
[1, 0] ]);
return dGdya,dGdyb
# ------------------------------------------------------------
def bcs(ya,yb):
return np.array([ ya[0] + 1.,
yb[0] - 1. ]);
# ------------------------------------------------------------
def init(x):
return np.array([ 1.-x,
-1. ]);
# ------------------------------------------------------------
options = struct()
# options include abstol, reltol, singularterm, stats, vectorized, maxnewpts,slopeout,xint
options.abstol, options.reltol = 1e-8, 1e-7
options.fjacobian, options.bcjacobian = f_jacobian, bc_jacobian
options.nmax = 20000
solinit = bvpinit(np.linspace(-1,1,10),init)
with nostdout():
sol = bvp6c(ode,bcs,solinit,options)
xint = np.linspace(-1,1,100); Sxint,_ = deval(sol,xint)
# savemat('np_vector.mat', {'x':sol.x,'y':sol.y,'xint':xint,'Sxint':Sxint})
m_data = loadmat(port_path+'test_data/bvp_prob7.mat')
grid_diff = np.max(np.abs(sol.x-m_data['x']))
sol_diff = np.max(np.abs(sol.y-m_data['y']))
grid_eval = np.max(np.abs(xint-m_data['xint']))
sol_eval = np.max(np.abs(Sxint-m_data['Sxint']))
if flag==True:
# plt.plot(sol.x,sol.y[1],linewidth=2.0)
# # plt.axis([-1.1, 1.1, 0., 1.2])
# plt.title('Numerical Solution')
# plt.xlabel('x'); plt.ylabel('y'); plt.show()
assert grid_diff < 1e-10,"Difference in grids = %e"%grid_diff
assert sol_diff < 1e-10, "Difference in solutions = %e"%sol_diff
assert grid_eval < 1e-10, "Difference in xint = %e"%grid_eval
assert sol_eval < 1e-10, "Difference in Sxint = %e"%sol_eval
print("Difference in grids = %e"%grid_diff)
print("Difference in solutions = %e"%sol_diff)
print("Difference in xint = %e"%grid_eval)
print("Difference in Sxint = %e"%sol_eval)
print("bvp_prob7 passes\n\n\n")
return (grid_diff, sol_diff,grid_eval, sol_eval)