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from numpy import *
from matplotlib.pyplot import *
import sys
def solver(I, a, T, dt, theta):
"""Solve u'=-a*u, u(0)=I, for t in (0,T] with steps of dt."""
dt = float(dt) # avoid integer division
Nt = int(round(T/dt)) # no of time intervals
T = Nt*dt # adjust T to fit time step dt
u = zeros(Nt+1) # array of u[n] values
t = linspace(0, T, Nt+1) # time mesh
u[0] = I # assign initial condition
for n in range(0, Nt): # n=0,1,...,Nt-1
u[n+1] = (1 - (1-theta)*a*dt)/(1 + theta*dt*a)*u[n]
return u, t
def u_exact(t, I, a):
return I*exp(-a*t)
def explore(I, a, T, dt, theta=0.5):
"""
Run a case with the solver, compute error measure,
and plot the numerical and exact solutions (if makeplot=True).
"""
u, t = solver(I, a, T, dt, theta) # Numerical solution
u_e = u_exact(t, I, a)
e = u_e - u
E = sqrt(dt*sum(e**2))
figure() # create new plot
t_e = linspace(0, T, 1001) # very fine mesh for u_e
u_e = u_exact(t_e, I, a)
plot(t, u, 'r--o') # dashed red line with circles
plot(t_e, u_e, 'b-') # blue line for u_e
legend(['numerical', 'exact'])
xlabel('t')
ylabel('u')
title('Method: theta-rule, theta=%g, dt=%g' % (theta, dt))
theta2name = {0: 'FE', 1: 'BE', 0.5: 'CN'}
savefig('%s_%g.png' % (theta2name[theta], dt), dpi=150)
savefig('%s_%g.pdf' % (theta2name[theta], dt))
#show() # run in batch
return E
def define_command_line_options():
import argparse
parser = argparse.ArgumentParser()
parser.add_argument('--I', '--initial_condition', type=float,
default=1.0, help='initial condition, u(0)',
metavar='I')
parser.add_argument('--a', type=float,
default=1.0, help='coefficient in ODE',
metavar='a')
parser.add_argument('--T', '--stop_time', type=float,
default=1.0, help='end time of simulation',
metavar='T')
parser.add_argument('--dt', '--time_step_values', type=float,
default=[1.0], help='time step values',
metavar='dt', nargs='+', dest='dt_values')
return parser
def read_command_line():
parser = define_command_line_options()
args = parser.parse_args()
return args.I, args.a, args.T, args.dt_values
def main():
"""Conduct experiments with theta and dt values."""
I, a, T, dt_values = read_command_line()
for theta in 0, 0.5, 1:
for dt in dt_values:
E = explore(I, a, T, dt, theta)
print '%3.1f %6.2f: %12.3E' % (theta, dt, E)
if __name__ == '__main__':
main()