def test_plot_solver_linear_error():
N = 3
rd = _get_decay_rd(N)
tout = np.linspace(0, 3.0, 7)
y0 = [3.0, 1.0] * N
integr = run(rd, y0, tout)
Cref = _get_decay_Cref(N, y0, tout)
ax = plot_solver_linear_error(integr, Cref)
assert isinstance(ax, matplotlib.axes.Axes)
def integrate_rd(D=2e-3, t0=3., tend=7., x0=0.0, xend=1.0, mu=None, N=32,
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,
atol=1e-6, rtol=1e-6, efield=False, random_seed=42,
verbose=False, use_log2=False):
if random_seed:
np.random.seed(random_seed)
n = 1
mu = float(mu or x0)
tout = np.linspace(t0, tend, nt)
assert geom in 'fcs'
# 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)
mob = 0.3
# Initial conditions
y0 = {
'f': y0_flat_cb,
'c': y0_cylindrical_cb,
's': y0_spherical_cb
}[geom](x, logx)
# Setup the system
stoich_active = []
stoich_prod = []
k = []
assert not lrefl
assert not rrefl
rd = ReactionDiffusion(
n, stoich_active, stoich_prod, k, N,
D=[D],
z_chg=[1],
mobility=[mob],
x=x,
geom=geom,
logy=logy,
logt=logt,
logx=logx,
nstencil=nstencil,
lrefl=lrefl,
rrefl=rrefl,
use_log2=use_log2
)
if efield:
if geom != 'f':
raise ValueError("Only analytic sol. for flat drift implemented.")
rd.efield = efield_cb(rd.xcenters, logx)
# Analytic reference values
t = tout.copy().reshape((nt, 1))
Cref = np.repeat(y0[np.newaxis, :, np.newaxis], nt, axis=0)
if efield:
Cref += t.reshape((nt, 1, 1))*mob
# Run the integration
integr = run(rd, y0, tout, atol=atol, rtol=rtol,
with_jacobian=(not num_jacobian), method=method)
Cout, info = integr.Cout, integr.info
if verbose:
print(info)
def lin_err(i=slice(None), j=slice(None)):
return integr.Cout[i, :, j] - Cref[i, :, j]
rmsd = np.sum(lin_err()**2 / N, axis=1)**0.5
ave_rmsd_over_atol = np.average(rmsd, axis=0)/info['atol']
# Plot results
if plot:
import matplotlib.pyplot as plt
def _plot(y, c, ttl=None, apply_exp_on_y=False):
plt.plot(rd.xcenters, rd.expb(y) if apply_exp_on_y else y, c=c)
if N < 100:
plt.vlines(rd.x, 0, np.ones_like(rd.x)*max(y), linewidth=.1,
colors='gray')
plt.xlabel('x / m')
plt.ylabel('C / M')
if ttl:
plt.title(ttl)
for i in range(nt):
c = 1-tout[i]/tend
c = (1.0-c, .5-c/2, .5-c/2) # over time: dark red -> light red
plt.subplot(4, 1, 1)
_plot(Cout[i, :, 0], c, 'Simulation (N={})'.format(rd.N),
apply_exp_on_y=logy)
plt.subplot(4, 1, 2)
_plot(Cref[i, :, 0], c, 'Analytic', apply_exp_on_y=logy)
ax_err = plt.subplot(4, 1, 3)
plot_solver_linear_error(integr, Cref, ax_err, ti=i,
bi=slice(None),
color=c, fill=(i == 0))
plt.title('Linear rel error / Log abs. tol. (={})'.format(
info['atol']))
plt.subplot(4, 1, 4)
tspan = [tout[0], tout[-1]]
plt.plot(tout, rmsd[:, 0] / info['atol'], 'r')
plt.plot(tspan, [ave_rmsd_over_atol[0]]*2, 'r--')
plt.xlabel('Time / s')
plt.ylabel(r'$\sqrt{\langle E^2 \rangle} / atol$')
plt.tight_layout()
plt.show()
return tout, Cout, info, ave_rmsd_over_atol, rd
def integrate_rd(
tend=1.9, A0=4.2, B0=3.1, C0=1.4, nt=100, t0=0.0, kf=0.9, kb=0.23,
atol='1e-7,1e-6,1e-5', rtol='1e-6', integrator='scipy', method='bdf',
logy=False, logt=False, num_jac=False, plot=False, savefig='None',
splitplots=False, plotlogy=False, plotsymlogy=False, plotlogt=False,
scale_err=1.0, scaling=1.0, verbose=False):
"""
Runs the integration and (optionally) plots:
- Individual concentrations as function of time
- Reaction Quotient vs. time (with equilibrium constant as reference)
- Numerical error commited (with tolerance span plotted)
- Excess error committed (deviation outside tolerance span)
Concentrations (A0, B0, C0) are taken to be in "M" (molar),
kf in "M**-1 s**-1" and kb in "s**-1", t0 and tend in "s"
"""
rtol = float(rtol)
atol = list(map(float, atol.split(',')))
if len(atol) == 1:
atol = atol[0]
registry = SI_base_registry.copy()
registry['amount'] = 1.0/scaling*registry['amount']
registry['length'] = registry['length']/10 # decimetre
kf = kf/molar/second
kb = kb/second
rd = ReactionDiffusion.nondimensionalisation(
3, [[0, 1], [2]], [[2], [0, 1]], [kf, kb], logy=logy, logt=logt,
unit_registry=registry)
C0 = np.array([A0, B0, C0])*molar
if plotlogt:
eps = 1e-16
tout = np.logspace(np.log10(t0+eps), np.log10(tend+eps), nt)*second
else:
tout = np.linspace(t0, tend, nt)*second
integr = Integration(
rd, C0, tout, integrator=integrator, atol=atol, rtol=rtol,
with_jacobian=not num_jac, method=method)
Cout = integr.with_units('Cout')
yout, info = integr.yout, integr.info
try:
import mpmath
assert mpmath # silence pyflakes
except ImportError:
use_mpmath = False
else:
use_mpmath = True
time_unit = get_derived_unit(registry, 'time')
conc_unit = get_derived_unit(registry, 'concentration')
Cref = _get_Cref(
to_unitless(tout - tout[0], time_unit),
to_unitless(C0, conc_unit),
[to_unitless(kf, 1/time_unit/conc_unit),
to_unitless(kb, 1/time_unit)],
use_mpmath
).reshape((nt, 1, 3))*conc_unit
if verbose:
print(info)
if plot:
npltcols = 3 if splitplots else 1
import matplotlib.pyplot as plt
plt.figure(figsize=(18 if splitplots else 6, 10))
def subplot(row=0, idx=0, adapt_yscale=True, adapt_xscale=True,
span_all_x=False):
offset = idx if splitplots else 0
ax = plt.subplot(4, 1 if span_all_x else npltcols,
1 + row*npltcols + offset)
if adapt_yscale:
if plotlogy:
ax.set_yscale('log')
elif plotsymlogy:
ax.set_yscale('symlog')
if adapt_xscale and plotlogt:
ax.set_xscale('log')
return ax
tout_unitless = to_unitless(tout, second)
c = 'rgb'
for i, l in enumerate('ABC'):
# Plot solution trajectory for i:th species
ax_sol = subplot(0, i)
ax_sol.plot(tout_unitless, to_unitless(Cout[:, 0, i], molar),
label=l, color=c[i])
if splitplots:
# Plot relative error
ax_relerr = subplot(1, 1)
ax_relerr.plot(
tout_unitless, Cout[:, 0, i]/Cref[:, 0, i] - 1.0,
label=l, color=c[i])
ax_relerr.set_title("Relative error")
ax_relerr.legend(loc='best', prop={'size': 11})
# Plot absolute error
ax_abserr = subplot(1, 2)
ax_abserr.plot(tout_unitless, Cout[:, 0, i]-Cref[:, 0, i],
label=l, color=c[i])
ax_abserr.set_title("Absolute error")
ax_abserr.legend(loc='best', prop={'size': 11})
# Plot absolute error
linE = Cout[:, 0, i] - Cref[:, 0, i]
try:
atol_i = atol[i]
except:
atol_i = atol
wtol_i = (atol_i + rtol*yout[:, 0, i])*get_derived_unit(
rd.unit_registry, 'concentration')
if np.any(np.abs(linE/wtol_i) > 1000):
# Plot true curve in first plot when deviation is large enough
# to be seen visually
ax_sol.plot(tout_unitless, to_unitless(Cref[:, 0, i], molar),
label='true '+l, color=c[i], ls='--')
ax_err = subplot(2, i)
plot_solver_linear_error(integr, Cref, ax_err, si=i,
scale_err=1/wtol_i, color=c[i], label=l)
ax_excess = subplot(3, i, adapt_yscale=False)
plot_solver_linear_excess_error(integr, Cref, ax_excess,
si=i, color=c[i], label=l)
# Plot Reaction Quotient vs time
ax_q = subplot(1, span_all_x=False, adapt_yscale=False,
adapt_xscale=False)
Qnum = Cout[:, 0, 2]/(Cout[:, 0, 0]*Cout[:, 0, 1])
Qref = Cref[:, 0, 2]/(Cref[:, 0, 0]*Cref[:, 0, 1])
ax_q.plot(tout_unitless, to_unitless(Qnum, molar**-1),
label='Q', color=c[i])
if np.any(np.abs(Qnum/Qref-1) > 0.01):
# If more than 1% error in Q, plot the reference curve too
ax_q.plot(tout_unitless, to_unitless(Qref, molar**-1),
'--', label='Qref', color=c[i])
# Plot the
ax_q.plot((tout_unitless[0], tout_unitless[-1]),
[to_unitless(kf/kb, molar**-1)]*2,
'--k', label='K')
ax_q.set_xlabel('t')
ax_q.set_ylabel('[C]/([A][B]) / M**-1')
ax_q.set_title("Transient towards equilibrium")
ax_q.legend(loc='best', prop={'size': 11})
for i in range(npltcols):
subplot(0, i, adapt_yscale=False)
plt.title('Concentration vs. time')
plt.legend(loc='best', prop={'size': 11})
plt.xlabel('t')
plt.ylabel('[X]')
subplot(2, i, adapt_yscale=False)
plt.title('Absolute error in [{}](t) / wtol'.format('ABC'[i]))
plt.legend(loc='best')
plt.xlabel('t')
ttl = '|E_i[{0}]|/(atol_i + rtol*(y0_i+yf_i)/2'
plt.ylabel(ttl.format('ABC'[i]))
plt.tight_layout()
subplot(3, i, adapt_yscale=False)
ttl = 'Excess error in [{}](t) / integrator linear error span'
plt.title(ttl.format(
'ABC'[i]))
plt.legend(loc='best')
plt.xlabel('t')
plt.ylabel('|E_excess[{0}]| / e_span'.format('ABC'[i]))
plt.tight_layout()
save_and_or_show_plot(savefig=savefig)
return yout, to_unitless(Cref, conc_unit), rd, info