PDE_FiniteDifference

# PDE_FiniteDifference - CE 3101 Fall 2011 Some Partial...

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Unformatted text preview: CE 3101 Fall 2011 Some Partial Differential Equations and Associated Finite Difference Schemes 1 Recall 1.1 Forward Difference ∂f ∂x ( x , y ) ≈ f ( x + Δx , y ) - f ( x , y ) Δx (1) ∂f ∂y ( x , y ) ≈ f ( x , y + Δy ) - f ( x , y ) Δy (2) 1.2 Central Difference ∂f ∂x ( x , y ) ≈ f ( x + Δx , y ) - f ( x- Δx , y ) 2 Δx (3) ∂f ∂y ( x , y ) ≈ f ( x , y + Δy ) - f ( x , y- Δy ) 2 Δy (4) ∂ 2 f ∂x 2 ( x , y ) ≈ f ( x + Δx , y ) - 2 f ( x , y ) + f ( x- Δx , y ) Δx 2 (5) ∂ 2 f ∂y 2 ( x , y ) ≈ f ( x , y + Δy ) - 2 f ( x , y ) + f ( x , y- Δy ) Δy 2 (6) 2 Case I Find φ ( x , y ) such that 1 , ∂ 2 φ ∂x 2 + ∂ 2 φ ∂y 2 = (7) We rewrite (7) using the finite difference approximations given by (5) and (6). φ ( x + Δx , y ) - 2 φ ( x , y ) + φ ( x- Δx , y ) Δx 2 + φ ( x , y + Δy ) - 2 φ ( x , y ) + φ ( x , y- Δy ) Δy 2 = (8) To simplify the presentation we introduce a local “compass-based” notation: φ o = φ ( x , y ) (9) φ w = φ ( x- Δx , y ) (10) φ e = φ ( x + Δx , y ) (11) φ n = φ ( x , y + Δy )...
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## This note was uploaded on 02/15/2012 for the course CIVIL 3101 taught by Professor Barnes during the Spring '11 term at Minnesota.

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PDE_FiniteDifference - CE 3101 Fall 2011 Some Partial...

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