Non-linear elliptic differential equation - python

[leonmate]

Hey guys,

I'm going to be honest and say I'm so stuck on this assignment - I really need help!
I've took on a third year computational physics course last year - turn your weaknesses into strengths someone told me.

Well, I failed and I'm back doing it again this year!

So, I just have to pass which means I need 30% in this assignment. But, I'm not getting there on my own - I've struggled to get to grips with the module - I got 20% in my last assignment :'(

Can someone help guide me through it?

So, I've already solved a similar problem, the one I did was:

u''(x) = f(x) = - exp(x) * (-2 + 2x + 5x² + x³)

from pylab import *
from scipy import interpolate
       
def EllipticSolver(Nx):
    dx = 1.0/Nx
    pts = linspace(-dx/2.0,1.+dx/2.0,Nx+2)
    soln = zeros(Nx+2)
    rhs = zeros(Nx+2)
    rhs = -exp(pts)*(-2.0 + 2.*pts + 5.*pts**2 + pts**3)
    soln = tridiagonal2(soln,rhs,dx)
    return pts[1:-1], soln[1:-1]
   
def tridiagonal2(dat,d,dx):
    N = len(dat)
    print(N)
    dx2 = 1.0/(dx*dx)
    a = zeros(N)
    b = zeros(N)
    c = zeros(N)
    a[:] = dx2
    c[:] = dx2
    b[:] = -2.0*dx2
   
    b[1] = -1.0*dx2
    b[N-2] = -3.0*dx2
  
    c[1] = c[1]/b[1]
    d[1] = d[1]/b[1]
    for j in range(2,N-1):
        c[j] = c[j]/(b[j] - c[j-1]*a[j])
        d[j] = (d[j] - d[j-1]*a[j])/(b[j] - c[j-1]*a[j])
       
    print(a)
    print(b)
    print(c)
    print(d)
  
    dat[N-2] = d[N-2]
    for j in range(N-3,0,-1):
        dat[j] = d[j] - c[j]*dat[j+1]
   
    return dat

x1, soln1 = EllipticSolver(100)
x2, soln2 = EllipticSolver(200)
x3, soln3 = EllipticSolver(400)

figure(1)
clf()
plot(x1,soln1,'r-')
plot(x2,soln2,'g-')

s2 = interpolate.interp1d(x2,soln2,'cubic')
soln2a = s2(x1)

s3 = interpolate.interp1d(x3,soln3,'cubic')
soln3a = s3(x2)

diff1 = soln1 - soln2a
diff2 = soln2 - soln3a

figure(2)
clf()
plot(x1,diff1,'r-')
plot(x2,4.*diff2,'g-')

Now I have to solve it for:

u'' + 500*exp(-100*x²) - u³ = 0

I guess I rearrange so:

u'' = u³ - 500*exp(-100*x²)

and then solve that

How do I solve it with an x and a u in the equation :O :(

If you've made it this far I love you and please help me, I have until next Wednesday to get this done and I can't do it on my own..... pleeeeeease help I just need 30 damn percent

I attached a copy of the assignmentLeon

 

[leonmate]

OK, so I've made a little progressFirst thing we have to do is find the finite-difference approximation for the u'' matrix for a staggered grid

So, Newton-Raphson method, discretise using finite-difference approximations of the derivatives

u'' = (u_i+1 + u_i-1 - 2u_i) / dx²

So, the first thing I'm going to have to do is set up some kind of array for u in order to whack it in a for loop and populate it using the above equation

I don't know what 'u' should be here, I could use an array of zeros, but I need some values in order to get something for u''

In the example above, i create the rhs array using the staggered grid I produced called pts

Do I need to make another staggered grid for u, and then use that to make a RHS array:

rhs = zeros(N)
rhs = u_i+1 + u_i-1 - 2u_i / dx²

That wouldn't quite work but something along those lines...