def simulate(which, boundary, delta_t):
'''
wrapper function for one plot. plots the different times in the so-called
array for the scheme ´which´ and boundary ´boundary´. The time step has the
size ´delta_t´.
'''
times = np.array([0.01, 0.04, 0.07])
how_many_times = np.asarray(times / delta_t, dtype=int)
xnum = 51
CFL = delta_t * (xnum + 1)**2
fig = plt.figure(figsize=(8, 5))
for i in range(3):
ival = get_ival(how_many_times[i], delta_t, xnum, boundary)
obj = simulation(ival, diffusionTable, which)
obj.simulate(ival.DiffFunc, 1.0)
plt.plot(ival.xvec,
obj.result,
'-o',
fillstyle='none',
label='$t=%.2f$, numerical' % times[i])
plt.plot(ival.xvec,
diffusionExact[boundary](ival.xvec, ival.N_T * ival.DELTA_T,
1.0),
'-',
label='$t=%.2f$, exact' % times[i])
plt.xlabel('$x$')
plt.ylabel('$u / [\\mathrm{a.u.}]$')
plt.title('%s, %s boundary, $\\mathsf{CFL} = %.3f$' % \
(translationTitle[which],boundary,CFL))
plt.grid()
plt.legend()
plt.tight_layout()
return fig
def error_plot(dtvec, dxvec, which, tmax):
'''
helper function to create the 3d-plot that is made for the error analysis.
'''
dt, dx = np.meshgrid(dtvec, dxvec)
res = np.zeros_like(dt)
for i, DT in enumerate(dtvec):
for j, DX in enumerate(dxvec):
ival = initialValues(
[None, 1.0,
int(1 / DX),
int(tmax / DT), DT, 'periodic'])
obj = simulation(ival, hopfTable, which)
obj.simulate(10 * ival.HopfFunc, 1.0)
res[i,j] = np.sqrt(np.sum((
10*advection_exact_periodic(ival.xvec,ival.N_T*ival.DELTA_T,1.0)- \
obj.result)**2))
return dt, dx, res
def error_plot(dtvec, dxvec, which, tmax):
'''
helper function to create the 3d-plot that is made for the error analysis.
Logarithmic scaling of all axes.
'''
dt, dx = np.meshgrid(dtvec, dxvec)
res = np.zeros_like(dt)
for i, DT in enumerate(dtvec):
for j, DX in enumerate(dxvec):
ival = initialValues(
[None, 1.0,
int(1 / DX),
int(tmax / DT), DT, 'reflecting'])
obj = simulation(ival, diffusionTable, which)
obj.simulate(ival.DiffFunc, 1.0)
res[i,j] = np.sqrt(np.sum((
diffusionExact['reflecting'](ival.xvec,ival.N_T*ival.DELTA_T,1.0)- \
obj.result)**2))
return np.log10(dt), np.log10(dx), np.log(res)
def make_sim(nmax):
ival.DiffStep = fourier_rect(ival.xvec,h=0.5,N=nmax) + \
ran_profile(ival.xvec,h=1.0,N=nmax) + 0.5
nfive = simulation(ival, diffusionTable, 'cr_n_D')
nfive.simulate(u_init, 1.0)
return nfive.result
ax.set_ylabel('$\\mathrm{log}(\\mathsf{TE})$')
ax.set_title(
'Truncation Error ($\\mathsf{TE}$) for the explicit {\\scshape Euler}-scheme; $\\mathsf{CLF} = 1/6$'
)
plt.grid()
plt.tight_layout()
fig_eu_e_16.savefig(prefix + 'eu_e_ERRORS16.pdf')
T = 1000
eps = 5e-3
DT = 1e-6
ival = initialValues([None, 1.0, 701, T, DT, 'periodic'], eps=eps)
u_init = eps / np.pi / ((ival.xvec - 0.5)**2 + eps**2)
# general check for the cr_n_Dfun-simulation.
if False:
cnsd = simulation(ival, diffusionTable, 'cr_n_D')
cnsd.simulate(eps / np.pi / ((ival.xvec - 0.5)**2 + eps**2), 1.0)
cnsd.plot_init(plot_default=False)
plt.plot(ival.xvec, cnsd.result, label='simulated')
plt.plot(ival.xvec,
exact_dfun(ival.xvec, T * DT, ival.DiffStep, x0=0.5),
label='exact')
plt.grid()
plt.xlabel('$x$')
plt.ylabel('$u(x,t)/\\mathrm{[a.u.]}$')
plt.title('Step profile, mathematical comparison')
plt.legend()
plt.tight_layout()
cnsd.fig.savefig(prefix + 'cr_n_D_COMPARE.pdf')
from helpers import fourier_rect, many_steps, ran_profile
'lax_w': lax_wendroff,
'lax_w_h': lax_wendroff_hopfeq
}
if False:
# dummy if, only used to not execute this part which was used to generate
# the plots for the hopf equation
a = 0.1
dt = 0.001
maxslope_anti = np.sqrt(np.exp(1)) * a
my_T = (int(maxslope_anti / dt) - 1) * dt
ival = initialValues(
[None, 1.0, 301,
int(maxslope_anti / dt) - 1, dt, 'periodic'])
up = simulation(ival, hopfTable, 'lax_w_h')
u_i = +np.exp(-(ival.xvec - 0.5)**2 / 2 / a**2) + 1
up.simulate(u_i, 1.0)
up.plot_init('$t=0$')
up.plot_final('$t=%.3f$' % my_T, 'Breakdown: $a = %.3f$' % a)
#up.fig.savefig('hopfBreakdown2.pdf')
plt.show()
# some human-readable output definitely is nice.
schemeNames = {
'up': 'upwind scheme',
'down': 'downwind scheme',
'ex_c': 'explicit centered scheme',
'im_c': 'implicit centered scheme',
'lax_f': '{\scshape Lax Friedrichs} scheme',
'lax_w': '{\scshape Lax Wendroff} scheme'