Differential Equations Lecture Work Solutions 314

Differential Equations Lecture Work Solutions 314 - 1 Apply...

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Unformatted text preview: 1. Apply the ADI scheme to the 2-D heat equation and find un+1 at the internal grid points in the mesh shown in figure 60 for rx = ry = 2. The initial conditions are x along y = 0 3∆x y un = 1 − along x = 0 2∆y un = 0 everywhere else un = 1 − and the boundary conditions remain fixed at their initial values. Step 1: rx n+1/2 n+1/2 n+1/2 n+1/2 − unj = ui+1 j − 2ui j + ui−1 j ui j i 2 =1 ry 2 + unj +1 − 2unj + unj −1 i i i =1 Step 2: n+1/2 +1 unj − ui j i = rx 2 n+1/2 n+1/2 ui+1 j − 2ui j n+1/2 + ui−1 j =1 ry 2 + +1 +1 +1 unj +1 − 2unj + unj −1 i i i =1 For y = 0, bottom boundary, the index j = 1, un1 = 1 − i x 3∆x 2 3 1 un 1 = 3 3 n and u4 1 are not needed un 1 = 2 un 1 1 For x = 0, left boundary i = 1 un j = 1 − 1 y 2∆y 1 2 is not needed un 2 = 1 un 3 1 un j = un3 = 0, given 4 i u0 2 = u0 2 = 0 2 3 314 ...
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This note was uploaded on 12/22/2011 for the course MAP 2302 taught by Professor Bell,d during the Fall '08 term at UNF.

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