def test_integrated_conc(params):
geom, logx = params
N = 8192
x0, xend = 0.11, 1.37
x = np.linspace(x0, xend, N+1)
rd = ReactionDiffusion(1, [], [], [], D=[0], N=N,
x=np.log(x) if logx else x, geom=geom, logx=logx)
xc = rd.expb(rd.xcenters) if logx else rd.xcenters
y = xc*np.exp(-xc)
def primitive(t):
if geom == 'f':
return -(t+1)*np.exp(-t)
elif geom == 'c':
return 2*np.exp(-t)*np.pi*(-2 - 2*t - t**2)
elif geom == 's':
return 4*np.exp(-t)*np.pi*(-6 - 6*t - 3*t**2 - t**3)
else:
raise NotImplementedError
res = rd.integrated_conc(y)
ref = (primitive(xend) - primitive(x0))
assert abs(res - ref) < 1e-8
def integrate_rd(N=64, geom='f', nspecies=1, nstencil=3,
D=2e-3, t0=3.0, tend=7., x0=0.0, xend=1.0, center=None,
nt=42, logt=False, logy=False, logx=False,
random=False, p=0, a=0.2,
linterpol=False, rinterpol=False, ilu_limit=5.0,
n_jac_diags=-1, num_jacobian=False,
method='bdf', integrator='cvode', iter_type='undecided',
linear_solver='default',
atol=1e-8, rtol=1e-10,
efield=False, random_seed=42, mobility=0.01,
plot=False, savefig='None', verbose=False, yscale='linear',
vline_limit=100, use_log2=False, Dexpr='[D]*nspecies', check_conserv=False
): # remember: anayltic_N_scaling.main kwargs
# Example:
# python3 analytic_diffusion.py --plot --Dexpr "D*np.exp(10*(x[:-1]+np.diff(x)/2))"
if t0 == 0.0:
raise ValueError("t0==0 => Dirac delta function C0 profile.")
if random_seed:
np.random.seed(random_seed)
# decay = (nspecies > 1)
# n = 2 if decay else 1
center = float(center or x0)
tout = np.linspace(t0, tend, nt)
assert geom in 'fcs'
analytic = {
'f': flat_analytic,
'c': cylindrical_analytic,
's': spherical_analytic
}[geom]
# Setup the grid
logx0 = math.log(x0) if logx else None
logxend = math.log(xend) if logx else None
if logx and use_log2:
logx0 /= math.log(2)
logxend /= math.log(2)
_x0 = logx0 if logx else x0
_xend = logxend if logx else xend
x = np.linspace(_x0, _xend, N+1)
if random:
x += (np.random.random(N+1)-0.5)*(_xend-_x0)/(N+2)
def _k(si):
return (si+p)*math.log(a+1)
k = [_k(i+1) for i in range(nspecies-1)]
rd = ReactionDiffusion(
nspecies,
[[i] for i in range(nspecies-1)],
[[i+1] for i in range(nspecies-1)],
k,
N,
D=eval(Dexpr),
z_chg=[1]*nspecies,
mobility=[mobility]*nspecies,
x=x,
geom=geom,
logy=logy,
logt=logt,
logx=logx,
nstencil=nstencil,
lrefl=not linterpol,
rrefl=not rinterpol,
ilu_limit=ilu_limit,
n_jac_diags=n_jac_diags,
use_log2=use_log2
)
if efield:
if geom != 'f':
raise ValueError("Only analytic sol. for flat drift implemented.")
rd.efield = _efield_cb(rd.xcenters)
# Calc initial conditions / analytic reference values
t = tout.copy().reshape((nt, 1))
yref = analytic(rd.xcenters, t, D, center, x0, xend,
-mobility if efield else 0, logy, logx, use_log2).reshape(nt, N, 1)
if nspecies > 1:
from batemaneq import bateman_parent
bateman_out = np.array(bateman_parent(k, tout)).T
terminal = (1 - np.sum(bateman_out, axis=1)).reshape((nt, 1))
bateman_out = np.concatenate((bateman_out, terminal), axis=1).reshape(
(nt, 1, nspecies))
if logy:
yref = yref + rd.logb(bateman_out)
else:
yref = yref * bateman_out
# Run the integration
integr = run(rd, yref[0, ...], tout, atol=atol, rtol=rtol,
with_jacobian=(not num_jacobian), method=method,
iter_type=iter_type, linear_solver=linear_solver,
C0_is_log=logy, integrator=integrator)
info = integr.info
if logy:
def lin_err(i, j):
linref = rd.expb(yref[i, :, j])
linerr = rd.expb(integr.yout[i, :, j])-linref
linatol = np.average(yref[i, :, j])
linrtol = linatol
return linerr/(linrtol*np.abs(linref)+linatol)
if logy:
rmsd = np.sum(lin_err(slice(None), slice(None))**2 / N, axis=1)**0.5
else:
rmsd = np.sum((yref-integr.yout)**2 / N, axis=1)**0.5
ave_rmsd_over_atol = np.average(rmsd, axis=0)/atol
if verbose:
# Print statistics
from pprint import pprint
pprint(info)
pprint(ave_rmsd_over_atol)
# Plot results
if plot:
import matplotlib.pyplot as plt
plt.figure(figsize=(6, 10))
# colors: (0.5, 0.5, 0.5), (0.5, 0.5, 1), ...
base_colors = list(product([.5, 1], repeat=3))[1:-1]
def color(ci, ti):
return np.array(base_colors[ci % len(base_colors)])*tout[ti]/tend
for ti in range(nt):
plt.subplot(4, 1, 1)
for si in range(nspecies):
plt.plot(rd.xcenters, integr.Cout[ti, :, si], c=color(si, ti),
label=None if ti < nt - 1 else rd.substance_names[si])
plt.subplot(4, 1, 2)
for si in range(nspecies):
plt.plot(rd.xcenters, rd.expb(yref[ti, :, si]) if logy
else yref[ti, :, si], c=color(si, ti))
plt.subplot(4, 1, 3)
if logy:
for si in range(nspecies):
plt.plot(rd.xcenters, lin_err(ti, si)/atol,
c=color(si, ti))
else:
for si in range(nspecies):
plt.plot(
rd.xcenters,
(yref[ti, :, si] - integr.yout[ti, :, si])/atol,
c=color(si, ti))
if N < vline_limit:
for idx in range(1, 4):
plt.subplot(4, 1, idx)
for bi in range(N):
plt.axvline(rd.x[bi], color='gray')
plt.subplot(4, 1, 1)
plt.title('Simulation (N={})'.format(rd.N))
plt.xlabel('x / m')
plt.ylabel('C / M')
plt.gca().set_yscale(yscale)
plt.legend()
plt.subplot(4, 1, 2)
plt.title('Analytic solution')
plt.gca().set_yscale(yscale)
plt.subplot(4, 1, 3)
plt.title('Linear rel. error / Abs. tol. (={})'.format(atol))
plt.subplot(4, 1, 4)
plt.title('RMS error vs. time'.format(atol))
tspan = [tout[0], tout[-1]]
for si in range(nspecies):
plt.plot(tout, rmsd[:, si] / atol, c=color(si, -1))
plt.plot(tspan, [ave_rmsd_over_atol[si]]*2,
c=color(si, -1), ls='--')
plt.xlabel('Time / s')
plt.ylabel(r'$\sqrt{\langle E^2 \rangle} / atol$')
plt.tight_layout()
save_and_or_show_plot(savefig=savefig)
if check_conserv:
tot_amount = np.zeros(tout.size)
for ti in range(tout.size):
for si in range(nspecies):
tot_amount[ti] += rd.integrated_conc(integr.yout[ti, :, si])
if plot:
plt.plot(tout, tot_amount)
plt.show()
assert np.allclose(tot_amount[0], tot_amount[1:])
return tout, integr.yout, info, ave_rmsd_over_atol, rd, rmsd
def integrate_rd(D=-3e-1, t0=0.0, tend=7., x0=0.1, xend=1.0, N=1024,
base=0.5, offset=0.25, nt=25, geom='f',
logt=False, logy=False, logx=False, random=False,
nstencil=3, lrefl=False, rrefl=False,
num_jacobian=False, method='bdf', plot=False,
savefig='None', atol=1e-6, rtol=1e-6, random_seed=42,
surf_chg=(0.0, 0.0), sigma_q=101, sigma_skew=0.5,
verbose=False, eps_rel=80.10, use_log2=False):
"""
A negative D (diffusion coefficent) denotes:
mobility := -D
D := 0
A positive D calculates mobility from Einstein-Smoluchowski relation
"""
assert 0 <= base and base <= 1
assert 0 <= offset and offset <= 1
if random_seed:
np.random.seed(random_seed)
n = 2
if D < 0:
mobility = -D
D = 0
else:
mobility = electrical_mobility_from_D(D, 1, 298.15)
print(D, mobility)
# Setup the grid
logb = (lambda arg: log(arg)/log(2)) if use_log2 else log
_x0 = logb(x0) if logx else x0
_xend = logb(xend) if logx else xend
x = np.linspace(_x0, _xend, N+1)
if random:
x += (np.random.random(N+1)-0.5)*(_xend-_x0)/(N+2)
# Setup the system
stoich_active = []
stoich_prod = []
k = []
rd = ReactionDiffusion(
n, stoich_active, stoich_prod, k, N,
D=[D, D],
z_chg=[1, -1],
mobility=[mobility, -mobility],
x=x,
geom=geom,
logy=logy,
logt=logt,
logx=logx,
nstencil=nstencil,
lrefl=lrefl,
rrefl=rrefl,
auto_efield=True,
surf_chg=surf_chg,
eps_rel=eps_rel, # water at 20 deg C
faraday_const=1,
vacuum_permittivity=1,
use_log2=use_log2
)
# Initial conditions
sigma = (xend-x0)/sigma_q
sigma = [(1-sigma_skew)*sigma, sigma_skew*sigma]
y0 = np.vstack(pair_of_gaussians(
rd.xcenters, [base+offset, base-offset], sigma, logy, logx, geom, use_log2)).transpose()
if logy:
y0 = sigm(y0)
if plot:
# Plot initial E-field
import matplotlib.pyplot as plt
plt.figure(figsize=(6, 10))
rd.calc_efield((rd.expb(y0) if logy else y0).flatten())
plt.subplot(4, 1, 3)
plt.plot(rd.xcenters, rd.efield, label="E at t=t0")
plt.plot(rd.xcenters, rd.xcenters*0, label="0")
# Run the integration
tout = np.linspace(t0, tend, nt)
integr = run(rd, y0, tout,
atol=atol, rtol=rtol, sigm_damp=True,
C0_is_log=logy,
with_jacobian=(not num_jacobian), method=method)
Cout = integr.Cout
if verbose:
print(integr.info)
# Plot results
if plot:
def _plot(y, ttl=None, **kwargs):
plt.plot(rd.xcenters, y, **kwargs)
plt.xlabel((('log_%s({})' % ('2' if use_log2 else 'e')) if logx else '{}').format('x / m'))
plt.ylabel('C / M')
if ttl:
plt.title(ttl)
for i in range(nt):
plt.subplot(4, 1, 1)
c = 1-tout[i]/tend
c = (1.0-c, .5-c/2, .5-c/2)
_plot(Cout[i, :, 0], 'Simulation (N={})'.format(rd.N),
c=c, label='$z_A=1$' if i == nt-1 else None)
_plot(Cout[i, :, 1], c=c[::-1],
label='$z_B=-1$' if i == nt-1 else None)
plt.legend()
plt.subplot(4, 1, 2)
delta_y = Cout[i, :, 0] - Cout[i, :, 1]
_plot(delta_y, 'Diff'.format(rd.N),
c=[c[2], c[0], c[1]],
label='A-B (positive excess)' if i == nt-1 else None)
plt.legend(loc='best')
plt.xlabel("$x~/~m$")
plt.ylabel(r'Concentration / M')
ylim = plt.gca().get_ylim()
if N < 100:
plt.vlines(rd.x, ylim[0], ylim[1],
linewidth=1.0, alpha=0.2, colors='gray')
plt.subplot(4, 1, 3)
plt.plot(rd.xcenters, rd.efield, label="E at t=tend")
plt.xlabel("$x~/~m$")
plt.ylabel("$E~/~V\cdot m^{-1}$")
plt.legend()
for i in range(3):
plt.subplot(4, 1, i+1)
ylim = plt.gca().get_ylim()
for d in (-1, 1):
center_loc = [x0+(base+d*offset)*(xend-x0)]*2
plt.plot(rd.logb(center_loc) if logx else center_loc,
ylim, '--k')
plt.subplot(4, 1, 4)
for i in range(n):
amount = [rd.integrated_conc(Cout[j, :, i]) for j in range(nt)]
plt.plot(tout, amount, c=c[::(1, -1)[i]], label=chr(ord('A')+i))
plt.xlabel('Time / s')
plt.ylabel('Amount / mol')
plt.legend(loc='best')
plt.tight_layout()
save_and_or_show_plot(savefig=savefig)
return tout, Cout, integr.info, rd
