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

๐๐ข∗

๐๐ก∗

๐๐ข∗

+ ๐ข∗ = ๐

๐๐ฅ∗

๐2๐ข∗

๐๐ฅ∗2

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

๐๐ข ๐๐ข ๐2๐ข

๐๐ก + ๐ข ๐๐ฅ = ๐๐ฅ2

b) Verify that is an analytical solution

2 sinh ๐ฅ

๐ข = − cosh ๐ฅ − ๐−๐ก

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 ๏ฃx๏ฃ9 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 ๐ข๐ for the first term of the non-linear term of the Burger’s eqn. viz (denoting j = spatial location, n = time iteration level)

๐ข๐+1 = ๐ข๐ − โ๐ก

๐ข๐(๐ข๐ − ๐ข๐ ) + โ๐ก

(๐ข๐ − 2๐ข๐ + ๐ข๐ )

๐ ๐

2โ๐ฅ ๐

๐+1

๐−1

(โ๐ฅ)2

๐+1

๐ ๐−1

Note, here the Fourier number is

โ๐ก

๐น๐ =

(โ๐ฅ)2

And the FTCS explicit scheme is stable when Fo ๏ฃ1/2

d) Use MacCormack’s Explicit finite difference discretization scheme to numerically solve the non-dimensional Burger’s eqn. of a) on the domain -9 ๏ฃx๏ฃ9 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

๐๐ข ๐๐ธ ๐2๐ข

๐๐ก + ๐๐ฅ = ๐๐ฅ2

๐ธ =

1 ๐ข2

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

Predictor:

๐ข๐ = ๐ข๐ − โ๐ก (๐ธ๐ − ๐ธ๐) + โ๐ก

(๐ข๐ − 2๐ข๐ + ๐ข๐ )

๐ ๐

โ๐ฅ

๐+๐ ๐

(โ๐ฅ)2

๐+1

๐ ๐−1

Corrector:

๐ข๐ = ๐ข๐ − โ๐ก (๐ธ๐ − ๐ธ๐) + โ๐ก

(๐ข๐ − 2๐ข๐ + ๐ข๐ )

๐ ๐

โ๐ฅ ๐+๐ ๐

(โ๐ฅ)2

๐+1

๐ ๐−1

Then

๐ข๐+1 = 0.5(๐ข๐ + ๐ข๐)

๐ ๐ ๐

Note the above scheme is stable according to

1

โ๐ก ≤ 1 2

โ๐ฅ + (โ๐ฅ)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 ๏ฃx๏ฃ9 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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