from pylab import *

# =======================================================
# Advection equation: du/dt + v*du/dx = 0 
# with perdiodic boundary condition
# 3 explicit schemes : FTCS, Lax-Friedrichs, Upwind
# =======================================================


# equation properties --------------------------
L  = 1.0
v  = 1.0


# analytical solution --------------------------
def uexacte(t,x):
    return ??


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


# time-sampling --------------------------------
tmax = ??
cfl  =  ??
dt = ??
nt = int(tmax/dt)+1
print '--------------------------------------------------'
print "CFL=%5.2f tmax =%7.4f --> %i iterations en temps"%(cfl,(nt+1)*dt,nt)
print '--------------------------------------------------'

 
# initial condition ----------------------------
t=0. 
u = uexacte(0.,x)
plot(x,u,'-o',label='Initial')


#  Time loop ----------------------------------
for i in range(1,nt+1):
    
    # scheme including boundary conditions
    u = ??

    # 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
            p = plot(x,uexacte(t,x))
            plot(x,u,'--o',color=p[0].get_color())


# Plot last time step ---------------------------
p = plot(x,uexacte(t,x),label='Analytical')
plot(x,u,'--o',color=p[0].get_color(),label='Numerical')


# Plot setup ------------------------------------
if (not intermediate):
    ylim([0,1.2*max(uexacte(t,x))])
plot(x, ones(nx),color='k',ls='--')
plot(x,-ones(nx),color='k',ls='--')
xlabel('x')
ylabel('u(t,x)')
ylim([-1.1,1.3])
title('Explicit scheme for the advection equation with CFL=%5.2f'%cfl)
legend(loc='upper center',ncol=3)
show()