from pylab import *
# =======================================================
# Burgers equation: du/dt + u*du/dx = 0 
# with perdiodic boundary conditions and Lax-Friedrich
# scheme
# =======================================================


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


# Intial condition -----------------------------
def uinit(x):
    return ??


# 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
print '--------------------------------------------------'
print "CFL=%5.2f tmax =%7.4f --> %i iterations en temps"%(cfl,nt*dt,nt)
print '--------------------------------------------------'

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


#  Time loop ----------------------------------
for i in range(1,nt+1):
    
    # Lax-Friedrichs
    u = ???

    # upwind
    # u = ??

    # update time
    t=t+dt

    # Optional: plot some intermediate time steps
    intermediate = 1
    if (intermediate):
        dtplot = 0.2
        if (i % int(dtplot/dt) == 0):
            print "t=%5.2f"%t
            plot(x,u,'--o')

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


# Plot setup ------------------------------------
plot(x, ones(nx),color='k',ls='--')
plot(x,zeros(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 Burgers equation with CFL=%5.2f'%cfl)
legend(loc='upper center',ncol=3)
show()