def get_tight_coupling_index(x, k, rec_info, bg_params, x_end_rec):
""" Here, x and k are assumed to be an n_k * n_x grid of values """
n_k = x.shape[0]
n_x = x.shape[1]
#We only evaluate the first column of the x grid and copy the result to
#the other columns since it is independent of k
tau_p = np.zeros(x.shape)
tau_p[0] = rec_info.get_tau_primed(x[0])
tau_p[1:] = np.array((n_k - 1) * [tau_p[0]])
#This is evaluated on a n_x * n_k grid
ck_byH_ptau_p = s_o_l * k / (tau_p * time_mod.get_H_p(x, bg_params))
#Again, we find the index just for the first tau array.
ind = np.searchsorted(np.abs(tau_p[0])[::-1], 10)
ind = n_x - ind
#This also is independent of k, so we do this step first
if x_end_rec < x[0, ind]:
ind = np.searchsorted(x[0, :], x_end_rec)
#Here, there is a k dependence so we make a list comprehension that loops
#through all ks
ind = np.array([ind if ck_byH_ptau_p[i, ind] <= 0.1 else np.searchsorted(ck_byH_p_tau_p[i, :], 0.1) for i in xrange(n_k)])
#Ind should now be an n_k array
return ind
def __init__(self,
bg_params,
a_start=None,
a_end=None,
a_npoints=None,
x_array=None,
tau_init=0):
if x_array is None:
self.x_array = np.linspace(np.log(a_start), np.log(a_end), a_npoints)
else:
self.x_array = x_array
self.a_array = np.exp(self.x_array)
self.bg_params = bg_params
if tau_init != 0:
raise NotImplementedError('tau_init must currently be zero')
self.tau_init = tau_init
#Precompute info for splining
self.ne_array = compute_ne(self.x_array, self.bg_params)
self.tau_der_array = -self.ne_array * s_o_l * sigma_T * np.exp(self.x_array) / time_mod.get_H_p(self.x_array, self.bg_params)
#Reversing the array because we're starting at today and
#integrating backwards
self.tau_array = np.append(np.array([self.tau_init]), cumtrapz(self.tau_der_array[::-1], self.x_array[::-1]))[::-1]
#Typically, this won't matter, and it avoids log issues
self.tau_array[-1] = self.tau_array[-2]
self.g_array = - self.tau_der_array * np.exp(-self.tau_array)
#Set up splines
self.ne_spline = interpolation.cubic_spline(self.x_array, np.log(self.ne_array))
self.tau_spline = interpolation.cubic_spline(self.x_array, np.log(self.tau_array))
self.tau_der_spline = interpolation.cubic_spline(self.x_array, np.log(-self.tau_der_array))
self.g_spline = interpolation.cubic_spline(self.x_array, self.g_array)
#Spline the second derivative of the log of g
self.g2_spline = interpolation.cubic_spline(self.x_array, self.g_spline)
def evolve_boltzmann(x, k, bg_params, rec_info, eta_info, lmax, lmax_nu, x_end_rec):
""" x and k should be n_k * n_x grids """
n_k = x.shape[0]
n_x = x.shape[1]
ind = get_tight_coupling_index(x, k, rec_info, bg_params, x_end_rec)
#Don't necessarily have to do this for every x, could evaluate for one k
#and copy, but this step is only done once, so very little to save here
divH_p = 1 / time_mod.get_H_p(x, bg_params)
divtau_p = 1 / rec_info.get_tau_primed(x)
ck = s_o_l * k
a = np.exp(x)
H_0 = bg_params.H_0
omega_nu = bg_params.omega_nu
omega_r = bg_params.omega_r
divf_nu = (omega_nu + omega_r) / omega_nu
theta = np.zeros((lmax + 1, n_k, n_x))
theta_p = np.zeros((lmax + 1, n_k, n_x))
n = np.zeros((lmax_nu + 1, n_k, n_x))
#Set up initial conditions
phi = np.ones((n_k, n_x))
delta = 1.5 * phi
delta_b = delta
v = 0.5 * ck * divH_p * phi
v_b = v
theta[0] = 0.5 * phi
theta[1] = -ck * divH_p * phi / 6
n[0] = 0.5 * phi
n[1] = -ck * divH_p / 6 * phi
n[2] = - ck ** 2 * a ** 2 * phi / (12 * H_0 ** 2 * omega_nu * (2.5 * divf_nu + 1))
for l in xrange(3, lmax_nu + 1):
n[l] = ck * divH_p / (2 * l + 1) * n[l-1]
#Just some handy indices (valid after tight coupling)
theta_p_start = lmax + 3
n_start = 2 * lmax + 4
n_end = 2 * lmax + lmax_nu + 5
# print ind
# print k[:, 0]
# sys.exit()
#Solve the actual equations
#TODO - could this be done without the for loop?
for k_i in xrange(n_k):
print k_i
#if k_i == 5:
# sys.exit()
#First evolve until tight coupling is no longer valid
x_tight_coupling = x[k_i, :ind[k_i]]
y0 = np.array([])
#This can also be done better, I am sure
for el in (phi[k_i, 0], v_b[k_i, 0], theta[:2, k_i, 0], n[:, k_i, 0], delta_b[k_i, 0], v[k_i, 0], delta[k_i, 0]):
y0 = np.append(y0, el)
#Do the integration
print 'tight_coupling'
out = ode_mod.Output(nsave=-1, force_stepsize=True, xarr=x_tight_coupling)
tc_solver = ode_mod.OdeInt(y0, x_tight_coupling[0], x_tight_coupling[-1], atol=1e-3, rtol=1e-3, h1=0.01, hmin=0.0, out=out, derivs=einstein_boltzmann_tight_coupling(k[k_i, 0], rec_info, eta_info, lmax, lmax_nu, bg_params))
tc_solver.integrate()
res = tc_solver.out.ysave
print 'nok', tc_solver.nok
print 'nbad', tc_solver.nbad
phi[k_i, 0:ind[k_i]] = res[0, :]
v_b[k_i, 0:ind[k_i]] = res[1, :]
theta[:2, k_i, 0:ind[k_i]] = res[2:4, :]
n[:, k_i, 0:ind[k_i]] = res[4:lmax_nu + 5, :]
delta_b[k_i, 0:ind[k_i]] = res[lmax_nu + 5, :]
v[k_i, 0:ind[k_i]] = res[lmax_nu + 6, :]
delta[k_i, 0:ind[k_i]] = res[lmax_nu + 7, :]
#Set up the derived quantities during tight coupling. Alternative here
#is to wait to after the loop, but since ind will be k-dependent, this
#means copying more than strictly necessary
theta[2, k_i, 0:ind[k_i]] = -8 * ck[k_i, 0:ind[k_i]] * divH_p[k_i, 0:ind[k_i]] * divtau_p[k_i, 0:ind[k_i]] / 15 * theta[1, k_i, 0:ind[k_i]]
for l in xrange(3, lmax + 1):
theta[l, k_i, 0:ind[k_i]] = - l * ck[k_i, 0:ind[k_i]] / (2 * l + 1) * divH_p[k_i, 0:ind[k_i]] * divtau_p[k_i, 0:ind[k_i]] * theta[l-1, k_i, 0:ind[k_i]]
theta_p[0, k_i, 0:ind[k_i]] = 1.25 * theta[2, k_i, 0:ind[k_i]]
theta_p[1, k_i, 0:ind[k_i]] = -0.25 * ck[k_i, 0:ind[k_i]] * divH_p[k_i, 0:ind[k_i]] * divtau_p[k_i, 0:ind[k_i]] * theta[2, k_i, 0:ind[k_i]]
theta_p[2, k_i, 0:ind[k_i]] = 0.25 * theta[2, k_i, 0:ind[k_i]]
for l in xrange(3, lmax + 1):
theta_p[l, k_i, 0:ind[k_i]] = - l / (2 * l + 1) * ck[k_i, 0:ind[k_i]] * divH_p[k_i, 0:ind[k_i]] * divtau_p[k_i, 0:ind[k_i]] * theta_p[l-1, k_i, 0:ind[k_i]]
#Now evolve after tight coupling
x_after_tc = x[k_i, ind[k_i] - 1:]
y0 = np.array([])
#Use last step of tight coupling as initial conditions for current step
for el in (phi[k_i, ind[k_i]-1], v_b[k_i, ind[k_i]-1], theta[:,k_i, ind[k_i]-1], theta_p[:,k_i, ind[k_i]-1], n[:,k_i, ind[k_i]-1], delta_b[k_i, ind[k_i]-1], v[k_i, ind[k_i]-1], delta[k_i, ind[k_i]-1]):
y0 = np.append(y0, el)
print 'post'
print 'y0', y0
out = ode_mod.Output(nsave=-1, force_stepsize=True, xarr=x_after_tc)
tc_solver = ode_mod.OdeInt(y0, x_after_tc[0], x_after_tc[-1], atol=1e-3, rtol=1e-3, h1=0.01, hmin=0.0, out=out, derivs=einstein_boltzmann(k[k_i, 0], rec_info, eta_info, lmax, lmax_nu, bg_params), stepper='sie')
tc_solver.integrate()
res = tc_solver.out.ysave
print 'nok', tc_solver.nok
print 'nbad', tc_solver.nbad
phi[k_i, ind[k_i]-1:] = res[0, :]
v_b[k_i, ind[k_i]-1:] = res[1, :]
theta[:,k_i, ind[k_i]-1:] = res[2:theta_p_start, :]
theta_p[:, k_i, ind[k_i] - 1:] = res[theta_p_start:n_start, :]
n[:, k_i, ind[k_i] - 1:] = res[n_start:n_end, :]
delta_b[k_i, ind[k_i] - 1:] = res[n_end, :]
v[k_i, ind[k_i] - 1:] = res[n_end + 1, :]
delta[k_i, ind[k_i] - 1:] = res[n_end + 2, :]
np.savetxt('k%02d_phi.dat' % k_i, phi[k_i, :])
np.savetxt('k%02d_v_b.dat' % k_i, v_b[k_i, :])
for l in range(lmax + 1):
np.savetxt('k%02d_theta%02d.dat' % (k_i, l), theta[l, k_i])
np.savetxt('k%02d_theta_p%02d.dat' % (k_i, l), theta_p[l, k_i])
for l in range(lmax_nu + 1):
np.savetxt('k%02d_n%02d.dat' % (k_i, l), n[l, k_i])
np.savetxt('k%02d_delta_b.dat' % k_i, delta_b[k_i, :])
np.savetxt('k%02d_v.dat' % k_i, v[k_i, :])
np.savetxt('k%02d_delta.dat' % k_i, delta[k_i, :])
np.savetxt('k%02d_ckdivH_p.dat' % k_i, (ck * divH_p)[k_i, :])
np.savetxt('k%02d_12H0divk2a2.dat' % k_i, (12 * H_0 ** 2 / ck ** 2 / a ** 2)[k_i, :])
np.savetxt('k%02d_psi.dat' % k_i, (-phi - 12 * H_0 ** 2 / ck ** 2 / a ** 2 * (omega_r * theta[2] + omega_nu * n[2]))[k_i])
return (phi, v_b, theta, theta_p, n, delta_b, v, delta)
def jacobian(self, x, y):
diva = np.exp(-x)
omega_r = self.bg_params.omega_r
omega_b = self.bg_params.omega_b
omega_nu = self.bg_params.omega_nu
omega_m = self.bg_params.omega_m
tau_p = self.rec_info.get_tau_primed(x)
divH_p = 1 / time_mod.get_H_p(x, self.bg_params)
l_array = self.l_array
delta_l_2 = self.delta_l_2
diveta = 1 / self.eta_info.get_eta(x)
div3 = self.div3
ck = self.ck
ckdivH_p = ck * divH_p
divck = self.divck
H_0 = self.bg_params.H_0
lmax = self.lmax
lmax_nu = self.lmax_nu
theta_start = 2
theta_p_start = self.theta_p_start
n_start = self.n_start
n_end = self.n_end
R = 4 * omega_r * div3 * diva / omega_b
numvars = len(y)
dpi = np.zeros(numvars)
dpi[theta_start + 2] = 1
dpi[theta_p_start] = 1
dpi[theta_p_start+2] = 1
dpsi = np.zeros(numvars)
dpsi[0] = -1
dpsi[theta_start + 2] = -12 * H_0 ** 2 * diva ** 2 * divck ** 2 * omega_r
dpsi[n_start + 2] = -12 * H_0 ** 2 * diva ** 2 * divck ** 2 * omega_nu
res = np.zeros((numvars, numvars))
#Dtheta
res[0, 0] = - ckdivH_p ** 2 * div3
res[0, theta_start] = 2 * (H_0 * divH_p * diva) ** 2 * omega_r
res[0, n_start] = 2 * (H_0 * divH_p * diva) ** 2 * omega_nu
res[0, n_end] = 0.5 * diva * (H_0 * divH_p) ** 2 * omega_m
res[0, n_end + 2] = 0.5 * diva * (H_0 * divH_p) ** 2 * omega_b
res[0, :] += dpsi
#Dv_b
res[1, 1] = -1 + tau_p * R
res[1, theta_start + 1] = 3 * tau_p * R
res[1, :] -= ckdivH_p * dpsi
#Dtheta0
res[theta_start, theta_start + 1] = -ckdivH_p
res[theta_start, :] -= res[0, :]
#Dtheta1
res[theta_start + 1, 1] = tau_p * div3
res[theta_start + 1, theta_start] = ckdivH_p * div3
res[theta_start + 1, theta_start + 1] = tau_p
res[theta_start + 1, theta_start + 2] = -2 * ckdivH_p * div3
res[theta_start + 1, :] -= ckdivH_p * div3 * dpsi
#Dtheta2
res[theta_start + 2, theta_start + 1] = 0.4 * ckdivH_p
res[theta_start + 2, theta_start + 2] = tau_p
res[theta_start + 2, theta_start + 3] = - 0.6 * ckdivH_p
res[theta_start + 2, :] -= 0.1 * tau_p * dpi
#Dthetal
base_array = np.array([l_array * ckdivH_p / (2 * l_array + 1), [tau_p] * len(l_array), - (l_array + 1) * ckdivH_p / (2 * l_array + 1)]).T
base_matrix = np.array([[base_array[l, i-l+1] if abs(l-i) < 2 else 0 for i in xrange(2, lmax + 1)] for l in xrange(3, lmax)])
res[theta_start + 3:theta_p_start - 1, theta_start + 2:theta_p_start] = base_matrix
#Dthetalmax
res[theta_p_start-1, theta_p_start - 2] = -ckdivH_p
res[theta_p_start - 1, theta_p_start - 1] = -(lmax + 1) * divH_p * diveta + tau_p
#Dtheta_p0
res[theta_p_start, theta_p_start] = tau_p
res[theta_p_start, theta_p_start + 1] = -ckdivH_p
res[theta_p_start, :] -= 0.5 * tau_p * dpi
#Dtheta_pl
base_matrix = np.array([[base_array[l, i-l+1] if abs(l-i) < 2 else 0 for i in xrange(0, lmax+1)] for l in xrange(1, lmax)])
res[theta_p_start + 1:n_start - 1, theta_p_start:n_start] = base_matrix
#For theta2 we have the pi contribution
res[theta_p_start + 2, :] -= 0.1 * tau_p * dpi
#Dtheta_plmax
res[n_start - 1, n_start - 2] = ckdivH_p
res[n_start - 1, n_start - 1] = - (lmax + 1) * divH_p * diveta + tau_p
#N0
res[n_start, n_start + 1] = -ckdivH_p
res[n_start, :] -= res[0, :]
#N1
res[n_start + 1, n_start] = ckdivH_p * div3
res[n_start + 1, n_start + 2] = -2 * ckdivH_p * div3
res[n_start + 1, :] += ckdivH_p * div3 * dpsi
#Nl
base_array = np.array([l_array * ckdivH_p / (2 * l_array + 1), np.zeros(len(l_array)), - (l_array + 1) * ckdivH_p / (2 * l_array + 1)]).T
base_matrix = np.array([[base_array[l, i-l+1] if abs(l-i) == 1 else 0 for i in xrange(1, lmax_nu + 1)] for l in xrange(2, lmax_nu)])
res[n_start + 2:n_end - 1, n_start + 1:n_end] = base_matrix
#Nlmax
res[n_end-1, n_end-2] = ckdivH_p
res[n_end-1, n_end-1] = - (lmax_nu + 1) * divH_p * diveta
#delta_b
res[n_end, 1] = ckdivH_p
res[n_end, :] -= 3 * res[0, :]
#v
res[n_end + 1, n_end + 1] = -1
res[n_end + 1, :] -= ckdivH_p * dpsi
#delta
res[n_end + 2, n_end + 1] = ckdivH_p
res[n_end + 2, :] -= 3 * res[0, :]
return res
def __call__(self, x, y):
""" Returns dy/dx, where y is the vector of all the Einstein-Boltzmann equations """
#Set up useful aliases
res = np.zeros(len(y))
diva = np.exp(-x)
omega_r = self.bg_params.omega_r
omega_b = self.bg_params.omega_b
omega_nu = self.bg_params.omega_nu
omega_m = self.bg_params.omega_m
#print x
tau_p = self.rec_info.get_tau_primed(x)
divH_p = 1 / time_mod.get_H_p(x, self.bg_params)
l_array = self.l_array
delta_l_2 = self.delta_l_2
diveta = 1 / self.eta_info.get_eta(x)
div3 = self.div3
ck = self.ck
H_0 = self.bg_params.H_0
lmax = self.lmax
lmax_nu = self.lmax_nu
theta_p_start = self.theta_p_start
n_start = self.n_start
n_end = self.n_end
phi = y[0]
v_b = y[1]
theta = y[2:theta_p_start]
theta_p = y[theta_p_start:n_start]
n = y[n_start:n_end]
delta_b = y[n_end]
v = y[n_end + 1]
delta = y[n_end + 2]
#Begin calculations
R = 4 * omega_r * div3 * diva / omega_b
PI = theta[2] + theta_p[0] + theta_p[2]
psi = -phi - 12 * H_0 ** 2 * (omega_r * theta[2] + omega_nu * n[2]) * diva ** 2 / ck ** 2
#phi
res[0] = psi - ck ** 2 * phi * divH_p ** 2 * div3 + H_0 ** 2 * 0.5 * divH_p ** 2 * (omega_m * delta * diva + omega_b * diva * delta_b + 4 * omega_r * diva ** 2 * theta[0] + 4 * omega_nu * diva ** 2 * n[0])
#vb
res[1] = -v_b - ck * divH_p * psi + tau_p * R * (3 * theta[1] + v_b)
# print 'res', res[1]
# print 'vb', v_b
# print 'ck', ck
# print 'divH_p', divH_p
# print 'psi', psi
# print 'tau_p', tau_p
# print 'R', R
# print 'threethetaplusvb', 3 * theta[1] + v_b
# print 'theta1', theta[1]
#theta0
res[2] = -ck * divH_p * theta[1] - res[0]
#theta1
res[3] = ck * div3 * divH_p * theta[0] - 2 * ck * div3 * divH_p * theta[2] + ck * div3 * divH_p * psi + tau_p * (theta[1] + div3 * v_b)
# print 'theta0', theta[0]
# print 'theta2', theta[2]
# print 'theta1plusonethirdvb', theta[1] + div3 * v_b
# print 'dphi', res[0]
# print 'ckdivHtheta1', ck*divH_p * theta[1]
#theta2-lmax-1
res[4:theta_p_start-1] = ck * divH_p * l_array[2:lmax] / (2 * l_array[2:lmax] + 1) * theta[1:lmax-1] - (l_array[2:lmax] + 1) * ck * divH_p / (2 * l_array[2:lmax] + 1) * theta[3:lmax+1] + tau_p * (theta[2:lmax] - 0.1 * PI * delta_l_2[2:lmax])
# print 'dtheta2', res[4]
# print 'theta3', theta[3]
# print 'thetap2', theta_p[2]
# print 'thetap0', theta_p[0]
# print 'pi', PI
#thetalmax
res[theta_p_start - 1] = ck * divH_p * theta[lmax - 1] - (lmax + 1) * divH_p * diveta * theta[lmax] + tau_p * theta[lmax]
#Theta_p
res[theta_p_start] = -ck * divH_p * theta_p[1] + tau_p * (theta_p[0] - 0.5 * PI)
res[theta_p_start+1:n_start-1] = ck * divH_p * l_array[1:lmax] / (2 * l_array[1:lmax] + 1) * theta_p[0:lmax-1] - (l_array[1:lmax] + 1) * ck * divH_p / (2 * l_array[1:lmax] + 1) * theta_p[2:lmax+1] + tau_p * (theta_p[1:lmax] - 0.1 * PI * delta_l_2[1:lmax])
res[n_start - 1] = ck * divH_p * theta_p[lmax - 1] - (lmax + 1) * divH_p * diveta * theta_p[lmax] + tau_p * theta_p[lmax]
#Neutrino multipoles
res[n_start] = -ck * divH_p * n[1] - res[0]
res[n_start + 1] = ck * divH_p * div3 * n[0] - 2 * ck * divH_p * div3 * n[2] + ck * divH_p * div3 * psi
res[n_start+2:n_end-1] = ck * divH_p * l_array[2:lmax_nu] / (2 * l_array[2:lmax_nu] + 1) * n[1:lmax_nu-1] - (l_array[2:lmax_nu] + 1) * ck * divH_p * (2 * l_array[2:lmax_nu] + 1) * n[3:lmax_nu + 1]
res[n_end - 1] = ck * divH_p * n[lmax_nu-1] - (lmax_nu + 1) * n[lmax_nu] * divH_p * diveta
#db, v, d
res[n_end] = ck * divH_p * v_b - 3 * res[0]
res[n_end + 1] = -v - ck * divH_p * psi
res[n_end + 2] = ck * divH_p * v - 3 * res[0]
return res
def __call__(self, x, y):
""" Returns dy/dx, where y is the vector of all the Einstein-Boltzmann equations, during tight coupling """
#Set up useful aliases
res = np.zeros(len(y))
diva = np.exp(-x)
tau_p = self.rec_info.get_tau_primed(x)
tau_dp = self.rec_info.get_tau_double_primed(x)
divH_p = 1 / time_mod.get_H_p(x, self.bg_params)
dH_p = time_mod.get_dH_p(x, self.bg_params)
diveta = 1 / self.eta_info.get_eta(x)
n_start = self.n_start
n_end = self.n_end
lmax = self.lmax
ck = self.ck
lmax_nu = self.lmax_nu
k = self.k
H_0 = self.H_0
div3 = self.div3
delta_l_2 = self.delta_l_2
l_array = self.l_array
omega_r = self.omega_r
omega_b = self.omega_b
omega_nu = self.omega_nu
omega_m = self.omega_m
phi = y[0]
v_b = y[1]
theta = y[2:n_start]
n = y[n_start:n_end]
delta_b = y[n_end]
v = y[n_end + 1]
delta = y[n_end + 2]
theta_2 = -8 * ck * divH_p / (15 * tau_p) * theta[1]
#Start calculations
R = 4 * omega_r * div3 * diva / omega_b
psi = -phi - 12 * H_0 ** 2 * (omega_r * theta_2 + omega_nu * n[2]) * diva ** 2 / ck ** 2
#Phi
res[0] = psi - ck ** 2 * phi * divH_p ** 2 * div3 + H_0 ** 2 * 0.5 * divH_p ** 2 * (omega_m * delta * diva + omega_b * diva * delta_b + 4 * omega_r * diva ** 2 * theta[0] + 4 * omega_nu * diva ** 2 * n[0])
#Theta0
res[2] = -ck * divH_p * theta[1] - res[0]
q = (-((1 - 2*R) * tau_p + (1 + R) * tau_dp) * (3 * theta[1] + v_b) - ck * divH_p * psi + (1 - dH_p * divH_p) * ck * divH_p * (-theta[0] + 2 * theta_2) - ck * divH_p * res[2]) / ((1 + R) * tau_p + dH_p * divH_p - 1)
#vb
res[1] = (-v_b - ck * divH_p * psi + R * (q + ck * divH_p*(-theta[0] + 2 * theta_2) - ck * divH_p * psi)) / (1 + R)
# print 'vb', res[1]
#Theta1
res[3] = div3 * (q - res[1])
#Neutrino multipoles
res[n_start] = -ck * divH_p * n[1] - res[0]
res[n_start + 1] = ck * divH_p * div3 * n[0] - 2 * ck * divH_p * div3 * n[2] + ck * divH_p * div3 * psi
res[n_start+2:n_end-1] = ck * divH_p * l_array[2:lmax_nu] / (2 * l_array[2:lmax_nu] + 1) * n[1:lmax_nu-1] - (l_array[2:lmax_nu] + 1) * ck * divH_p * (2 * l_array[2:lmax_nu] + 1) * n[3:lmax_nu + 1]
res[n_end - 1] = ck * divH_p * n[lmax_nu-1] - (lmax_nu + 1) * n[lmax_nu] * divH_p * diveta
#db, v, d
res[n_end] = ck * divH_p * v_b - 3 * res[0]
res[n_end + 1] = -v - ck * divH_p * psi
res[n_end + 2] = ck * divH_p * v - 3 * res[0]
# print 'v', res[n_end + 1]
return res
