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Unformatted text preview: 2. Apply the BeamWarming scheme with Euler implicit time diﬀerencing to the linearized
Burgers’ equation on the computational grid given in Figure 61 (use c = 2, µ = 2, ∆x = 1)
and determine the steady state values of u at j = 2 and j = 3. the boundary conditions are
u n = 1,
1
u 1 = 0,
2 and the initial conditions are un = 4
4
u1 = 0
3 Do not use a computer to solve this problem.
un+1 − un
j
j
+ cux
∆t
Let ν = c n+1
j n+1 = µuxx j ∆t
∆t
and r = µ 2 then
∆x
∆x
ν n+1
+1
+1
+1
u
− un−1 + r un−1 − 2un+1 + un+1
un+1 = un −
j
j
j
j
j
j
2 j +1 or
− ν
ν
+1
+1
+ r un−1 + (2r + 1)un+1 +
− r un+1 = un
j
j
j
j
2
2 In our case c = 2, µ = 2, ∆x = 1 and so if we let ∆t = 1, then r = ν = 2. Using these
values in the above equation, we get
+1
+1
−3un−1 + 5un+1 − un+1 = un
j
j
j
j For n = 1 (don’t forget to employ the boundary conditions)
−3 · 1 + 5u2 − u2 = 0
2
3
−3u2 + 5u2 − 4
2
3 =0 The solution of this system of two equations is
u2 =
2 19
22 u2 =
3 29
22 Now go to the next time step n = 2
= 19
22 −3u3 + 5u3 − 4 =
2
3 29
22 −3 + 5u3 − u3
2
3 The solution of this system of two equations is
333 ...
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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.
 Fall '08
 BELL,D

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