Differential Equations Lecture Work Solutions 331

# Differential Equations Lecture Work Solutions 331 - 1 Apply...

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Unformatted text preview: 1. Apply the two-step Lax Wendroﬀ method to the PDE ∂u ∂F ∂3u + +u 3 = 0 ∂t ∂x ∂x where F = F (u). Develop the ﬁnal ﬁnite diﬀerence equations. Recall that for ut + Fx = µuxx we had Step 1: n n+1/2 uj − 1 2 un+1/2 j 1 ∆t 2 + un−1/2 j Step 2: + − ∆x Fjn /2 +1 Fjn 1/2 − n+1/2 uxx =µ n j +1/2 + uxx j −1/2 2 n+1/2 un+1 − un Fj +1/2 − Fj −1/2 j j + = µuxx ∆t ∆x n j In our case we have uuxxx instead of µuxx , so Step 1: n n+1/2 uj − 1 2 un+1/2 j 1 ∆t 2 + un−1/2 j Step 2: + − ∆x Fjn /2 +1 Fjn 1/2 − n+1/2 uuxxx + n+1/2 un+1 − un Fj +1/2 − Fj −1/2 j j + = uuxxx ∆t ∆x n j +1/2 + uuxxx 2 n =0 j We need uuxxx approximated to O (∆x2 ), one can show n uxxx j 4un+1 − 8un+1/2 + 8un−1/2 − 4un−1 j j j j = + O (∆x2 ) 3 ∆x Using this approximation shifted to j ± 1/2, we get n uxxx j +1/2 n uxxx j −1/2 4un+3/2 − 8un+1 + 8un − 4un−1/2 j j j j = 3 ∆x 4un+1/2 − 8un + 8un−1 − 4un−3/2 j j j j = 3 ∆x Substitute these in step 1 331 j −1/2 =0 ...
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