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Finite difference method derivation PDE

  1. Dec 19, 2016 #1
    1. The problem statement, all variables and given/known data
    Which algebraic expressions must be solved when you use finite difference approximation to solve the following Possion equation inside of the square :

    $$U_{xx} + U_{yy}=F(x,y)$$

    $$0<x<1$$ $$0<y<1$$
    Boundary condition $$U(x,y)=G(x,y)$$

    2. Relevant equations
    Central difference approximation
    $$U_{xx}=\frac{-2F(x)+F(x+h)+F(x-h)}{h^2}$$
    $$U_x=\frac{F(x+h)-F(x-h)}{2h}$$

    3. The attempt at a solution
    $$U_{xx}+U_{yy} = \frac{1}{h^2}[U(x+h,y)+U(x-h,y)-4U(x,y)+U(x,y+h)+U(x,y-h)]=F(x,y)$$
    $$U(x,y)=\frac{1}{4}[U(x+h,y)+U(x-h,y)+U(x,y-h)+U(x,y+h)-h^2F(x,y)]$$

    The books solution is
    $$\frac{1}{2}[U(i+1,j)+U(i-1,j)+U(i,j+1)+U(i,j-1)]-\frac{h^2}{4}F(x,y)$$

    I know why the book changed x,y to i,j... but I dont get why the fraction is 1/2 instead of 1/4 accross the entire equation.
     
  2. jcsd
  3. Dec 19, 2016 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    I think the book's expression is wrong; your computation is the standard one that appears in numerous textbooks (maybe not yours!) and in many web pages; just Google "laplacian + finite differences".
     
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