# Implicit method to solve the heat equation.
from mpl_toolkits.mplot3d import axes3d
import matplotlib.pyplot as pl
import numpy as np
import matplotlib.patches as mpatches

t0 = 0.0
t_final = 1.0
n_grid = 20
dt = t_final / n_grid
dx = 1.0 / n_grid
T = np.arange(t0, t_final+dt, dt)
X = np.arange(0, 1+dx, dx)
ax = pl.figure().add_subplot(111, projection='3d')
ax.set_xlabel('t')
ax.set_ylabel('x')
ax.set_zlabel('U')
ax.set_xlim([t0,t_final])
ax.set_ylim([0, 1])
Umax= .15
ax.set_zlim([0, Umax])

# Exact solution
def exact(t, x):
    return np.exp(-t)*np.sin(np.pi*x)/(np.pi*np.pi)
    
for t in T:
    for x in X:
        if t>t0:
            ax.plot([t-dt, t],[x, x],[exact(t-dt, x), exact(t, x)], 'b-', linewidth=0.5)
        if x>0.0:
            ax.plot([t, t],[x-dx, x],[exact(t, x-dx), exact(t,x)], 'b-', linewidth=0.5)

# Implicit method
r = dx/(dt*dt)/(np.pi*np.pi)
U = [[0.0 for _ in X] for _ in T]
A = [[0.0 for _ in X] for _ in X]
b = [0.0 for _ in X]
for i in range(len(X)):
    if i in [0, len(X)-1]:
        A[i][i] = 1.0
    else:
        A[i][i] = 1+2*r
        A[i][i-1] = -r
        A[i][i+1] = -r
for i in range(len(T)):
    if i==0:
        for j in range(len(X)):
            U[i][j] = exact(T[i], X[j])
            b[j] = U[i][j]
    else:
        b = np.linalg.lstsq(A, b)[0]
        b[0], b[-1] = 0.0, 0.0
        for j in range(len(X)):
            U[i][j] = b[j]

for i in range(len(T)):
    for j in range(len(X)):
        if i>0:
            ax.plot([T[i-1], T[i]],[X[j], X[j]],[U[i-1][j], U[i][j]], 'r-', linewidth=0.5)
        if j>0:
            ax.plot([T[i], T[i]],[X[j-1], X[j]],[U[i][j-1], U[i][j]], 'r-', linewidth=0.5)

bluePatch = mpatches.Patch(color='blue', label='Exact')
redPatch = mpatches.Patch(color='red', label='Implicit')
pl.legend(handles = [bluePatch, redPatch], loc=2)

pl.show()