Differential Equations Lecture Work Solutions 332

Differential Equations Lecture Work Solutions 332 - Step 1...

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Unformatted text preview: Step 1: n+1/2 uj − 1 2 un+1/2 + un−1/2 j j 1 ∆t 2 Fjn /2 − Fjn 1/2 +1 − + ∆x 4un+1/2 − 8un + 8un−1 − 4un−3/2 1 j j j j + un−1/2 j 3 2 ∆x 4un+3/2 − 8un+1 + 8un − 4un−1/2 1 j j j j + un+1/2 2j ∆x3 =0 1 Multiply through by ∆t and collect terms 2 Step 1: 1n 1 ∆t n+1/2 uj − uj +1/2 + un−1/2 + Fn − Fjn 1/2 j − 2 2 ∆x j +1/2 + ∆t un −8un + 8un−1 − 4un−3/2 j j j ∆x3 j −1/2 + un+1/2 4un+3/2 − 8un+1 + 8un j j j j =0 Step 2: un+1 − un + j j + ∆t n+1/2 n+1/2 Fj +1/2 − Fj −1/2 ∆x ∆t n u 4un+1 − 8un+1/2 + 8un−1/2 − 4un−1 = 0 j j j j 3j ∆x Another possibility is to include uuxxx term into F . Let 1 G = uuxx − u2 2x then ∂G 1 = ux uxx + uuxxx − 2ux uxx = uuxxx ∂x 2 So now the equation is ut + Fx + Gx = 0 or ut + (F + G)x = 0 For this equation we can use Lax Wendroff method as in class. 332 ...
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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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