finite_difference/newton_heat_conduction.py

import sympy as sp
import numpy as np
import matplotlib.pyplot as plt

def heat_conduction(T_0,T_s,k,dt,N=100,theta=0):
    T=np.zeros(N)
    T[0]=T_0
    for i in range(1,N):
        T[i]=(1-(1-theta)*dt*k)*T[i-1]/(1+theta*dt*k)-k*dt*T_s/(1+theta*dt*k)
    return T

T_0=100
T_s=20
k=0.1
dt=0.1e-1
N=1000
theta=[0,0.5,1,1.2]
T=[[] for i in range(len(theta))]
t=np.linspace(0,N*dt,N)
T_real=[(T_0-T_s)*np.exp(-k*i*dt)+T_s for i in range(N) ]
# print(T_real)
for i in range(len(theta)):
    T[i]=heat_conduction(T_0,T_s,k,dt,N=N,theta=theta[i])
    # print(T[-1])
plt.plot(t,T[0])
plt.plot(t,T[1])
plt.plot(t,T[2])    
plt.plot(t,T[3])
plt.plot(t,T_real)
plt.legend(["T_0","T_0.5","T_1","T_1.2","T_real"])
plt.savefig("newton_heat_conduction.png")