from pylab import *
# =======================================================
# Wave equation: d2u/dt2 - c^2*d2u/dx2 = 0 
# with perdiodic boundary condition
# and a CTCS scheme (Centered in Time Centered in Space)
# =======================================================


# equation properties --------------------------
L  = ??
c  = ??


# analytical solution --------------------------
def init(x):
    u    = ??
    dudt = ??
    return u,dudt


# space-sampling -------------------------------
nx = ??
dx = L/nx
x  = linspace(0,L-dx,nx)


# time-sampling --------------------------------
tmax = ??
cfl  = ??
dt   = ?? 
# Small change in dt to match exactly the last time 'tmax'
nt = int(tmax/dt); dt = tmax/nt; cfl = dt/dx*c
print '--------------------------------------------------'
print "CFL=%5.2f tmax =%7.4f --> %i iterations en temps"%(cfl,nt*dt,nt)
print '--------------------------------------------------'

 
# initial conditions ---------------------------
t=0. 
ua,dudt = init(x)
plot(x,ua,'-o',label='Initial')

# Fist time step
ub = ??

#  Time loop ----------------------------------
for i in range(1,nt+1):
    
    # CTCS
    ua = ??

    # Flip ua <-> ub
    ua, ub = ub, ua

    # update time
    t=t+dt

    # Optional: plot intermediate some time steps
    intermediate = 0
    if (intermediate):
        dtplot = 0.1
        if (i % int(dtplot/dt) == 0):
            print "t=%5.2f"%t
            plot(x,ub,'--o')

# Plot last time step ---------------------------
plot(x,ub,'--o',label='Numerical')


# Plot setup ------------------------------------
plot(x, ones(nx),color='k',ls='--')
plot(x,zeros(nx),color='k',ls='--')
xlabel('x')
ylabel('u(t,x)')
ylim([-.1,1.2])
title('Explicit scheme for the wave equation with CFL=%5.2f'%cfl)
legend(loc='upper center',ncol=3)
show()