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Advection Equation - Python

  1. Nov 26, 2014 #1
    1. The problem statement, all variables and given/known data

    What I'm trying to achieve is an advection equation that will run using 3 variables. I must confess I don't have a great grasp over this part of my course. Its pretty complicated!

    2. Relevant equations

    The code I have at the moment is:

    Code (Text):
    def SolveAdvectionEquation(a, b, t0, t1, Nx, Nt, v, method):
    dx = (float) (b-a)/Nx
    dt = (t1-t0)/Nt
    # We have Nx+1 points for Nx intervals, plus an extra ghost point.
    pts = linspace(a-dx,b,Nx+2)
    times = linspace(t0,t1,Nt+1)
    solndata = zeros((Nt+1,Nx+2))
    sigma2 = 2.*0.1**2
    solndata[0] = exp(-((pts-0.5)**2)/(sigma2))
    solndata[0] = ApplyBoundaryConditions(solndata[0])
    for tn in range(0,Nt):
    solndata[tn+1] = method(solndata[tn],dx,dt,v)
    solndata[tn+1] = ApplyBoundaryConditions(solndata[tn+1])
    return times, pts[1:], solndata[:,1:]
     


    Our variables:
    a and b are my limits for x. For my example this is -1 and 1
    likewise t0 and t1 are my limits for time
    Nt and Nx are the number of time and x intervals (there will be Nt+1 times from t=0 to t=T and Nx+1 points in the interval of physical interest, x=a to x=b)
    v is our Courant factor (dt/dx)
    the method I want to use is Euler, I have a function made for this too:

    Code (Text):
    def Euler(y0,dx,dt,v):

    y1 = zeros(len(y0))

    for i in range(1,len(y0)-1):

    y1 = y0 - v*dt*(y0[i+1]-y0[i-1])/(2.0*dx)

    return y1


    This code works for a single varible, I need it to work for three, I believe the structure is similar.

    The equations I want to solve are
    du/dt = v
    dv/dt = dw/dx
    dw/dt = dv/dx
    i.e., you have three variables (u, v, w) and the evolution equations for each of them depends on the other variables, or the spatial derivatives of the other variables. The assignment provides the initial conditions for each variable, since v = du/dt, and w = du/dx.

    3. The attempt at a solution

    I don't really know what to show you for this, I've got as far as I've shown above and really don't know where to go from there. I figure I need to alter the for loop at the bottom of the function?

    Really need some help here, this is actually a computational physics assignment but felt it was better suited for the computer science section. Let me know if it should really be in one of the physics forums.
     
    Last edited: Nov 26, 2014
  2. jcsd
  3. Nov 26, 2014 #2

    jedishrfu

    Staff: Mentor

    Not having really studied your code, I can give you some advice:

    Manually write down the steps of how you would solve it. For each step, think about the code you'd write. When you have a complicated step break it up into smaller steps. When you just don't know what to do invent a function to handle it with the input variables needed and the output to be generated. After a while, you get the hang of it.

    Note by creating placeholder functions for steps you can't implement yet, you make a program that can be run and debugged up to a certain point.
     
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