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 + x , y ) - f ( x , y ) x (1) f y ( x , y ) f ( x , y + y ) - f ( x , y ) y (2) 1.2 Central Difference f x ( x , y ) f ( x + x , y ) - f ( x- x , y ) 2 x (3) f y ( x , y ) f ( x , y + y ) - f ( x , y- y ) 2 y (4) 2 f x 2 ( x , y ) f ( x + x , y ) - 2 f ( x , y ) + f ( x- x , y ) x 2 (5) 2 f y 2 ( x , y ) f ( x , y + y ) - 2 f ( x , y ) + f ( x , y- y ) y 2 (6) 2 Case I Find ( x , y ) such that 1 , 2 x 2 + 2 y 2 = (7) We rewrite (7) using the finite difference approximations given by (5) and (6). ( x + x , y ) - 2 ( x , y ) + ( x- x , y ) x 2 + ( x , y + y ) - 2 ( x , y ) + ( x , y- y ) y 2 = (8) To simplify the presentation we introduce a local compass-based notation: o = ( x , y ) (9) w = ( x- x , y ) (10) e = ( x + x , y ) (11) n = ( x , y + y )...
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PDE_FiniteDifference - CE 3101 Fall 2011 Some Partial...

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