Legendre Gauss Lobatto collocation

Description

The lglpsmethods.py computes LGL nodes, quadrature weights and differentiation matrix, given the grid size or number of intervals (N). To use the code, import the LGL class from the lglpsmethods.py module.

Example

The following exercise demonstrates

1. use of differentiation matrix to compute numerical derivative of the sine function and compare with the analytical solution
2. integrate numerically using quadrature weights over the given interval and compare with analytical solution

python code

# plotting results
import matplotlib.pyplot as plt
# initial and final point
x0=0
xf=2*np.pi
# grid size
N=50
# LGL object which has the diff. matrix, quadrature weights and LGL collocation points
lgl=LGL(N+1)
# domain transformation [-1,1]->[x0,xf]
x=lgl.tau*(xf-x0)/2+(xf+x0)/2
# sample the function 
y=np.sin(x)
# compute derivative from differentiation matrix
y_dot=2/(xf-x0)*[email protected](N+1,1)
# plot function and its numerical and analytical derivative
plt.plot(x,y,label='sine')
plt.plot(x,y_dot,'-o',label='cosine (lgl)')
plt.plot(x,np.cos(x),label='cosine (analytical)')
#plt.plot(x,y_dot-np.cos(x))
plt.legend()
plt.grid()
plt.show()
print(lgl.wi.T@y)

output

quadrature: [[1.43226551e-17]]