Derivation of MWR coefficients - adv/diff example

[20]:

import numpy as np
import matplotlib.pyplot as plt
from sympy import *

Collocation method

[22]:

 a2, x, c, K = symbols('a2 x c K')

[23]:

xc      = 0.5 #collocation point
Ri      = c*((1-a2)+2*a2*xc) - K*2*a2
a2      = solve(Ri, a2)
a2

[23]:

[c/(2*K)]

Least Squares method

Try to compute the functional form of the coefficient a2 using the symbolic math package following the example given above.

[1]:

#a2, x, c, K = symbols('a2 x c K')
#Ri          = ???
#Wi          = diff(?,?)
#eq          = integrate(Wi*Ri,(x,0,1))
#a2          = solve(eq, a2)
#a2

[3]:

# hide
Ri          = c*((1-a2)+2*a2*x) - K*2*a2
Wi          = diff(Ri,a2)
a2          = solve(integrate(Wi*Ri,(x,0,1)),a2)
a2

[3]:

[6*K*c/(12*K**2 + c**2)]

Galerkin method

[27]:

a2, x, c, K = symbols('a2 x c K')
#Ri          = ???
#Wi          = ???
#a2          = ???
# a2

[28]:

# hide
Ri          = c*((1-a2)+2*a2*x) - K*2*a2
Wi          = (x**2-x);
a2          = solve(integrate(Wi*Ri,(x,0,1)),a2)
a2

[28]:

[c/(2*K)]

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