Picewise-linear-Rayleigh-Ritz/Rayleigh-Ritz Method.py at main · readikole/Picewise-linear-Rayleigh-Ritz · GitHub

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import numpy as np
import matplotlib.pyplot as plt
import matplotlib
matplotlib.rcParams['text.usetex'] = True


'''I use the piecewise linear Rayleigh-Ritz method to solve a second order  ordinary differential equation with a nonzero RHS
The d.e is of the form :  - d/dx(  p(x)dy/dx) + q(x)y = f(x).
The integration is performed on the interval 0<=x<=1'''

def rhs(x):
    '''This computes the rhs of the d.e.
    INPUT: the variable x'''
    return 2*(np.pi**2)*np.sin(np.pi*x)
N =100
x = np.linspace(0,1, N+1)

h = []          #intialize space step
h0 = x[1] - x[0]
h.append(h0)
for i in range(N-1):
        h.append(x[i+1] - x[i])
hN = x[N] - x[N-1]
h.append(hN)
def basis_function(x):
    '''Basis functions of the Rayleigh-Ritz method'''

    phi = np.zeros([N+1, N+1], dtype='float')                               #initialize basis function
    for j in range(N-1):
        for idx in range(N):
            if (0<=x[idx]<=x[j-1]):
                phi[j, idx] = 0
            elif (x[j-1]<x[idx]<=x[j]):
                phi[j, idx] = (x[idx] - x[j-1])/h[j-1]
            elif (x[j]<x[idx]<=x[j+1]):
                phi[j, idx] = ((x[j+1] - x[idx])/h[j])
            elif (x[j+1]<x[idx]<=1):
                phi[j, idx] = 0
    return phi
#basis_function(x)

#coefficient functions
def p(x):
    '''Defining the coefficient p in the d.e.'''
    return 1

def q(x):
    '''Defining the coefficient q in the d.e.'''
    return np.pi**2
#initialize vectors for computing integrals
Q1 = []
Q2 = []
Q3 = []
Q4 = []
Q5 = []
Q6 = []
#approximating the 6 integrals
for i in range(N-1):
    q1 = (h[i]/12)*(p(x[i]) + q(x[i+1]))
    Q1.append(q1)
    q2 = (h[i-1]/12)*(3*q(x[i]) + q(x[i-1]))
    Q2.append(q2)
    q3 = (h[i]/12)*(3*q(x[i]) + q(x[i+1]))
    Q3.append(q3)
    q4 = (h[i-1]/2)*(p(x[i]) + p(x[i-1]))
    Q4.append(q4)
    q5 = (h[i-1]/6)*(2*rhs(x[i]) + rhs(x[i-1]))
    Q5.append(q5)
    q6 = (h[i]/6)*(2*rhs(x[i]) + rhs(x[i+1]))
    Q6.append(q6)

#now compute  Q1,n , Q2,n , Q3,n , Q4,n Q4,n+1, Q5n, Q6,n

#q1n = (h[N]/12)*(p(x[N-1]) + q(x[N]))
#Q1.append(q1n)
q2n = (h[N-2]/12)*(3*q(x[N-1]) + q(x[N-2]))
Q2.append(q2n)
q3n = (h[N]/12)*(3*q(x[N-1]) + q(x[N]))
Q3.append(q3n)
q4n_last = (h[N-1]/2)*(p(x[N]) + p(x[N-1]))
q4n_second_last = (h[N-2]/2)*(p(x[N-1]) + p(x[N-2]))
Q4.append(q4n_second_last)
Q4.append(q4n_last)
q5n = (h[N-1]/6)*(2*rhs(x[N-1]) + rhs(x[N-2]))
Q5.append(q5n)
q6n = (h[N]/6)*(2*rhs(x[N-1]) + rhs(x[N]))
Q6.append(q6n)


alpha = np.zeros(N+1, dtype='float')
beta = np.zeros(N+1, dtype='float')
b = np.zeros(N+1, dtype='float')
a = np.zeros(N+1, dtype='float')
for i in range(N-1):
    alpha[i] = Q4[i] + Q4[i+1] + Q2[i] + Q3[i]
    beta[i] = Q1[i] - Q4[i+1]
    b[i] = Q5[i] + Q6[i]
alpha[N] = Q4[N-1] + Q4[N] + Q2[N-1] + Q3[N-1]
b[N] = Q5[N-1] + Q6[N-1]
a[0] = alpha[0]

#solve a symmetric tridiagonal linear system
zeta = np.zeros(N+1, dtype='float')
z = np.zeros(N+1, dtype='float')
c = np.zeros(N+1, dtype='float')                      #the weights
zeta[0] = beta[0]/alpha[0]
z[0] = b[0]/a[0]
for i in range(1, N-1):
    a[i] = alpha[i] - beta[i-1]*zeta[i-1]
    zeta[i] = beta[i]/a[i]
    z[i] = (b[i] - beta[i-1]*z[i-1])/a[i]
a[N] = alpha[N] - beta[N-1]*zeta[N-1]
z[N] = (b[N] - beta[N-1]*z[N-1])/a[N]

#indices for the weights
#indices must be reversed
indices = []
for i in range(N):
    a =  N-1-i
    indices.append(a)
#print(indices)
c[N] = z[N]

for i in indices:
    c[i] = z[i] - zeta[i]*c[i+1]
u = basis_function(x)
#plt.plot(u)
#plt.show()

#compute the error
error = np.zeros(N+1, dtype='float')
phi_new = np.dot(u, c)
for i in range(len(x)):
    error[i] = (abs(phi_new [i]-np.sin(np.pi*x[i])))

import pandas as pd
df = pd.DataFrame(c ,phi_new)
print(df)

#plotting
plt.grid()
plt.plot(x, phi_new,x,  np.sin(np.pi*x), 'r')
plt.legend(['Rayleigh-Ritz', 'Exact Solution'])
plt.xlabel(r'x')
plt.ylabel(r'$\phi(x)$')
plt.savefig('Results')
plt.show()

plt.grid()
plt.loglog(np.linspace(0, N, N+1),error, 'g')
plt.title(r'\textbf{Error growth of the integrator}')
plt.xlabel(r'x')
plt.ylabel(r'Error')
plt.savefig('Error')
plt.show()