class-notes_Poisson - Sec 9.1 Poisson Equation on a...

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Sec 9.1: Poisson Equation on a rectangular domain with Dirichlet Boundary conditions 1. Governing Equation: 2 u ∂x 2 + 2 u ∂y 2 = Δ u = f ( x, y ) , ( x, y ) [ a, b ] × [ c, d ] (1) 2. Boundary Conditions: u ( a, y ) = g 1 ( y ) , u ( b, y ) = g 2 ( y ) , u ( x, c ) = g 3 ( x ) , u ( x, d ) = g 4 ( x ) . (2) 3. Discretization: Spatial Coordinates: 2 endpoints in each direction x i = a + ( i - 1) δx = a + i b - a N , i = 1 , · · · , N + 1 y i = c + ( i - 1) δy = c + i d - c M , i = 1 , · · · , M + 1 (3) Unknowns and Data ( Notice the difference with the book: matrix storage convention): u ( j, k ) = u ( x k , y j ) , f ( j, k ) = f ( x k , y j ) , g 1 / 2 / 3 / 4 ( j ) : stored in u (4) Laplacian operator (Taylor Expansion) Δ u ( j, k ) = u ( j, k + 1) - 2 u ( j, k ) + u ( j, k - 1) ( δx ) 2 + u ( j + 1 , k ) - 2
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Unformatted text preview: ,k ) ( δy ) 2 + O (( δx ) 2 )+ O (( δy ) 2 ) (5) Approximate u ( j,k ) by w ( j,k ) which satisfies: w ( j,k + 1)-2 w ( j,k ) + w ( j,k-1) ( δx ) 2 + w ( j + 1 ,k )-2 w ( j,k ) + w ( j-1 ,k ) ( δy ) 2 = f ( j,k ) (6) When δx = δy = h , this becomes 4 w ( j,k )-w ( j,k + 1)-w ( j,k-1)-w ( j + 1 ,k )-w ( j-1 ,k ) =-h 2 f ( j,k ) (7) For the points near the boundaries, 1 or 2 of the values on the LHS of (7) are known. 1...
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  • Winter '08
  • LinZhi
  • Numerical Analysis, Green's function, Laplace operator, Laplace's equation, Poisson's equation, Dirichlet boundary conditions, matrix storage convention

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