Square Root Map for Falkner-Skan
import scipy as sp | |
from scipy.integrate import ode | |
from scipy.optimize import fsolve | |
from matplotlib import rc | |
rc('text', usetex=True) | |
tick_size = 'large' | |
label_size = 'xx-large' | |
rc('xtick', labelsize=8) | |
rc('ytick', labelsize=8) | |
rc('legend', fontsize=12) | |
rc('axes', labelsize=14) | |
fig_width_pt = 469.75502 | |
inches_per_pt = 1.0/72.27 | |
golden_mean = (sp.sqrt(5.0)-1.0)/2.0 | |
fig_width = fig_width_pt *inches_per_pt | |
fig_height = fig_width * golden_mean | |
fig_size = [fig_width, fig_height] | |
rc('figure', figsize=fig_size) | |
from matplotlib import pylab as plt | |
from fs import * | |
def int_fs(beta_0, beta, alpha, eta_inf): | |
args = sp.array([beta_0, beta, eta_inf]) | |
y0 = sp.array([0.0, 0.0, alpha]) | |
r = ode(fs.rates, fs.jac).set_integrator('zvode', rtol=1e-14, atol=1e-14, method='adams', with_jacobian=True, order=12) | |
t0 = 0.0 | |
t1 = 1.0 | |
nt = 200 | |
dt = t1 / float(nt) | |
r.set_initial_value(y0, t0).set_f_params(args).set_jac_params(args) | |
for i in xrange(nt): | |
r.integrate(r.t+dt) | |
return(r) | |
def int_tanh_fs(beta_0, beta, alpha, eta_inf, nt): | |
args = sp.array([beta_0, beta, eta_inf]) | |
y0 = sp.array([0.0, 0.0, alpha]) | |
r = ode(fs.rates, fs.jac).set_integrator('zvode', rtol=1e-14, atol=1e-14, method='adams', with_jacobian=True, order=12) | |
t0 = 0.0 | |
t1 = 1.0 | |
# instead of using a uniform point distribution, use a tanh | |
# mapping, which clusters the points near the wall where the | |
# variation is largest | |
# nt = 64 | |
t = sp.arctanh(sp.linspace(t0,t1,nt,endpoint=False)) | |
# t = t/t.max() | |
dt = sp.diff(t) | |
r.set_initial_value(y0, t0).set_f_params(args).set_jac_params(args) | |
for i in xrange(nt-1): | |
r.integrate(r.t+dt[i]) | |
return(r) | |
def int_sqrt_fs(beta_0, beta, alpha, eta_inf, nt): | |
args = sp.array([beta_0, beta, eta_inf]) | |
y0 = sp.array([0.0, 0.0, alpha]) | |
r = ode(fs.rates, fs.jac).set_integrator('zvode', rtol=2e-16, atol=2e-16, method='adams', with_jacobian=True, order=12) | |
t0 = 0.0 | |
t1 = 1.0 | |
# instead of using a uniform point distribution, use a square root | |
# mapping, which clusters the points near the wall where the | |
# variation is largest | |
# nt = 64 | |
x = sp.linspace(t0,t1,nt,endpoint=False) | |
t = y_of_x(eta_inf,x) | |
# t = t/t.max() | |
dt = sp.diff(t) | |
r.set_initial_value(y0, t0).set_f_params(args).set_jac_params(args) | |
for i in xrange(int(nt)-1): | |
r.integrate(r.t+dt[i]) | |
return(r) | |
def objective_func(x, beta_0, beta): | |
alpha = x[0] | |
eta_inf = x[1] | |
r = int_fs(beta_0, beta, alpha, eta_inf) | |
return(sp.array([(r.y[1]-1.0).real, (r.y[2]).real])) | |
def objective_func2(x, beta_0, beta, nt): | |
alpha = x[0] | |
eta_inf = 1.0 # x[1] | |
r = int_tanh_fs(beta_0, beta, alpha, eta_inf, nt) | |
return(sp.array([(r.y[1]-1.0).real, (r.y[2]).real])) | |
def objective_func3(x, beta_0, beta, nt): | |
alpha = x[0] | |
eta_inf = 1.0 # x[1] | |
r = int_sqrt_fs(beta_0, beta, alpha, eta_inf, nt) | |
return(sp.array([(r.y[1]-1.0).real, (r.y[2]).real])) | |
def extrap(y): | |
return(2*y[1]-y[0]) | |
def extrap2(y): | |
return(-(4*y[1]-y[0]-5*y[2])/2.0) | |
# sqrt mapping | |
def y_of_x(L,x): | |
return(L*x / sp.sqrt(1 - x**2)) | |
eta_inf = 1.0 | |
alpha = 1e-2 | |
beta_0 = 1.0 | |
x = sp.array([alpha, eta_inf]) | |
nnt = sp.unique(sp.logspace(3,12,73,base=2).astype(int)) | |
foo = sp.zeros((nnt.shape[0],2)) | |
deta0 = sp.zeros(nnt.shape[0]) | |
for i,nt in enumerate(nnt): | |
args = (beta_0, 0.0, int(nt)) | |
foo[i] = fsolve(objective_func3, x, args=args, xtol=2e-16) | |
if(i==0): | |
x[0] = alpha | |
elif(i==1): | |
x[0] = extrap(foo[i-1:i+1,0]) | |
else: | |
x[0] = extrap2(foo[i-2:i+1,0]) | |
x[1] = eta_inf | |
y = y_of_x(eta_inf, sp.linspace(0,1,nt,endpoint=False)) | |
foo[i,1] = y.max() | |
deta0[i] = y[1] - y[0] # initial point spacing off the wall | |
err = foo[:,0] - foo[foo.shape[0]-1,0] | |
plt.figure() | |
plt.loglog(nnt[0:nnt.shape[0]-1], abs(err[0:nnt.shape[0]-1]),label=r'$\eta=\frac{x}{\sqrt{1 - x^2}}$, $x \in [0,1)$') | |
plt.xlabel(r'$n$') | |
plt.ylabel(r'$\| \alpha_n - \alpha \|$') | |
plt.legend(loc=0) | |
plt.savefig("sqrt_map_cnvrg.png") | |
plt.figure() | |
plt.plot(nnt, foo[:,1],label=r'$\eta=\frac{x}{\sqrt{1 - x^2}}$, $x \in [0,1)$') | |
plt.xlabel(r'$n$') | |
plt.ylabel(r'$\eta_{\infty}$') | |
plt.legend(loc=0) | |
plt.savefig("eta_inf.png") | |
plt.figure() | |
plt.semilogy(nnt, deta0,label=r'$\eta=\frac{x}{\sqrt{1 - x^2}}$, $x \in [0,1)$') | |
plt.xlabel(r'$n$') | |
plt.ylabel(r'$\Delta \eta_{wall}$') | |
plt.legend(loc=0) | |
plt.savefig("deta0.png") | |
plt.show() |
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spectral convergence of the method
eta_inf vs. n
Delta eta at the wall vs. n