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2D Heat equation using FDM

  1. Apr 13, 2012 #1
    1. The problem statement, all variables and given/known data
    Given a steady-state heat transfer for a 100mx100m plate, to be discretized to 6 nodes, governed by a heat tranfer equation:

    U_xx + U_yy = 0

    Given dirichlet boundary conditions: U(0,y)=50, U(100,y) = 100,, neumann boundary: U_y(x,0)=0, U_y(x,100) = 0.

    Find the solution using FDM, applying a 5 point laplacian stencil. Use Gauss-Seidel method to solve for the system of equations arising from the FDM approximation. Use 0.01% relative error as stopping criterion.


    2. Relevant equations

    U_xx = (U_x-1,y -2U_x,y +U_x+1,y)/(Δx)^2
    U_yy = (U_x,y-1 -2U_x,y +U_x,y+1)/(Δy)^2

    Assume equal spacing of nodes.
    Use Δx=Δy for simplicity.

    3. The attempt at a solution

    Substituting the FDM approximations into the given PDE,

    U_x,y = .25*(U_x-1,j + U_x+1,j + U_x,j-1 + U_x,j+1)

    I'm not really sure about how I did the Gauss-Seidel part because what I got is that the temperature just linearly varies along X.

    U(0,y) = 50
    U(20,y) = 60
    U(40,y) = 70
    U(60,y) = 80
    U(80,y) = 90
    U(100,y) = 100

    I'm not really confident about the answer so I tried using Gaussian elimination and I got a different answer.
     
  2. jcsd
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