Consider the 1d Burger s eqn which can be written as

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1. Consider the 1-d Burger’s eqn. which can be written as

 

 𝜕𝑢∗

 

𝜕𝑡∗

 

 𝜕𝑢∗

 

+ 𝑢∗ = 𝜈

 

𝜕𝑥∗

 

 𝜕2𝑢∗

 

𝜕𝑥∗2 

 

a) Normalize the above eqn. using the parameters L (m) and  (m^2/sec) to arrive at the non-dimensional version as follows

 

𝜕𝑢 𝜕𝑢 𝜕2𝑢

 

𝜕𝑡 + 𝑢 𝜕𝑥 = 𝜕𝑥2

 

 

 

 

 

 

 

b) Verify that is an analytical solution

 

 2 sinh 𝑥

 

𝑢 = − cosh 𝑥 − 𝑒−𝑡

 

c) Use FTCS Explicit (Forward Time Centered Space) finite difference discretization scheme to numerically solve the non-dimensional Burger’s eqn. of a) on the domain -9 x9 subject to the initial conditions @ t = 0.1 given by the analytical solution of part b), and with boundary conditions

 

u=2 @ x = -9, u = -2 @ x = 9

 

Integrate to t = 1.0, using step sizes of x=0.2 for spatial discretization and t=0.01 for temporal discretization. Plot the FTCS solution and the error vs. x and explain the error characteristics.

 

 

 

Note for the non-linear term, when using FTCS explicit, use 𝑢𝑛 for the first term of the non-linear term of the Burger’s eqn. viz (denoting j = spatial location, n = time iteration level)

 

 𝑢𝑛+1 = 𝑢𝑛 −  ∆𝑡

 

𝑢𝑛(𝑢𝑛 − 𝑢𝑛   ) + ∆𝑡

 

(𝑢𝑛 − 2𝑢𝑛 + 𝑢𝑛  )

 

 𝑗 𝑗

 

2∆𝑥  𝑗 

 

𝑗+1

 

 

 

𝑗−1

 

 

 

 (∆𝑥)2

 

 𝑗+1

 

 𝑗 𝑗−1

 

 Note, here the Fourier number is

 

∆𝑡

 

𝐹𝑜 =

 

(∆𝑥)2

 

And the FTCS explicit scheme is stable when Fo 1/2

 

d) Use MacCormack’s Explicit finite difference discretization scheme to numerically solve the non-dimensional Burger’s eqn. of a) on the domain -9 x9 subject to the initial conditions @ t = 0.1 given by the analytical solution of part b), and with boundary conditions

 

 

 

u=2 @ x = -9, u = -2 @ x = 9

 

Integrate to t = 1.0, using step sizes of x=0.2 for spatial discretization and t=0.01 for temporal discretization. Plot the MacCormack’s solution and the error vs. x and explain the error characteristics.

 

 

 

Note, for MacCormack’s Explicit method, use the conservative form of Burger’s eqn. viz

 

𝜕𝑢 𝜕𝐸 𝜕2𝑢

 

𝜕𝑡 + 𝜕𝑥 = 𝜕𝑥2

 

 𝐸 =

 

 1 𝑢2

 

the predictor-corrector steps as follows (where j = spatial location, n = time iteration):

 

Predictor: 

 

𝑢𝑝 = 𝑢𝑛 − ∆𝑡 (𝐸𝑛   − 𝐸𝑛) + ∆𝑡

 

(𝑢𝑛 − 2𝑢𝑛 + 𝑢𝑛  )

 

 𝑗 𝑗

 

 ∆𝑥

 

 𝑗+𝑖 𝑗

 

 (∆𝑥)2

 

 𝑗+1

 

 𝑗 𝑗−1

 

 Corrector:

 

 𝑢𝑐 = 𝑢𝑝 − ∆𝑡 (𝐸𝑝   − 𝐸𝑝) + ∆𝑡

 

 (𝑢𝑝 − 2𝑢𝑝 + 𝑢𝑝  )

 

 𝑗 𝑗

 

 ∆𝑥 𝑗+𝑖 𝑗

 

 (∆𝑥)2

 

 𝑗+1

 

 𝑗 𝑗−1

 

 Then

 

 𝑢𝑛+1 = 0.5(𝑢𝑝 + 𝑢𝑐)

 

 𝑗 𝑗 𝑗

 

Note the above scheme is stable according to

 

1

 

∆𝑡 ≤ 1 2

 

∆𝑥 + (∆𝑥)2

 

e) Use FTCS Implicit (Forward Time Centered Space) finite difference discretization scheme to numerically solve the non-dimensional Burger’s eqn. of a) on the domain -9 x9 subject to the initial conditions @ t = 0.1 given by the analytical solution of part b), and with boundary conditions

 

u=2 @ x = -9, u = -2 @ x = 9

 

Integrate to t = 1.0, using step sizes of x=0.2 for spatial discretization and t=0.01 for temporal discretization. Plot the FTCS Implicit solution and the error vs. x and explain the error characteristics.

 

Note for FTCS implicit, use the same finite differencing as in FTCS explicit, except to make it implicit, use the (n+1)th time level for spatial differences and solve the resulting set of tridiagonal equations using a tridiagonal algorithm such as the Thomas Tridiagonal Algorithm which occurs frequently in numerical fluid flow and heat transfer courses.

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