from pylab import *

# =======================================================
# Diffusion equation: du/dt = D * d2u/dx2
# with Dirichlet bounday conditions
# FTCS (Forward Time Centered Space), explicit scheme 
# =======================================================


# equation properties --------------------------
L  = 1.0
D  = 1.0


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


# space-sampling -------------------------------
nx = ??
dx = L/(nx-1.)
x  = linspace(0,L,nx)


# time-sampling --------------------------------
tmax = ??
cfl  = ??
dt = ??
nt = int(tmax/dt)+1
print('--------------------------------------------------')
print("CFL={:5.2f} tmax ={:7.4f} --> {:d} iterations en temps".format(cfl,nt*dt,nt))
print('--------------------------------------------------')

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


#  Time loop ----------------------------------
for it in range(1,nt+1):

    # FTCS scheme for inner points (1<=i<=nx-2)
    u = ??

    # Boundary conditions (i=0 and i=nx-1)
    u = ??
    
    # update time
    t = t + dt 
    
    # Optional: plot intermediate some time steps
    intermediate = 0
    if (intermediate):
        dtplot = 0.1
        if (it % int(dtplot/dt) == 0):
            print("t={:5.2f}".format(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))])
xlabel('x')
ylabel('u(t,x)')
title('Explicit FTCS scheme for the heat equation with CFL={:5.2f}'.format(cfl))
legend(loc='upper center',ncol=3)
show()