logo Use SA10RAM to get 10%* Discount.
Order Now logo

Ask This Question To Be Solved By Our ExpertsGet A+ Grade Solution Guaranteed

expert
Anmol AroraGeneral article writing
(5/5)

966 Answers

Hire Me
expert
Timothy PriceEducation
(5/5)

569 Answers

Hire Me
expert
ISAAC SSEKISAMBUPolitical science
(/5)

604 Answers

Hire Me
expert
Bill BattershilllAccounting
(5/5)

769 Answers

Hire Me
MATLAB
(5/5)

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

INSTRUCTIONS TO CANDIDATES
ANSWER ALL QUESTIONS

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

 

 ๐œ•๐‘ข∗

 

๐œ•๐‘ก∗

 

 ๐œ•๐‘ข∗

 

+ ๐‘ข∗ = ๐œˆ

 

๐œ•๐‘ฅ∗

 

 ๐œ•2๐‘ข∗

 

๐œ•๐‘ฅ∗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

 

๐œ•๐‘ข ๐œ•๐‘ข ๐œ•2๐‘ข

 

๐œ•๐‘ก + ๐‘ข ๐œ•๐‘ฅ = ๐œ•๐‘ฅ2

 

 

 

 

 

 

 

b) Verify that is an analytical solution

 

 2 sinh ๐‘ฅ

 

๐‘ข = − cosh ๐‘ฅ − ๐‘’−๐‘ก

 

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

 

 ๐‘ข๐‘›+1 = ๐‘ข๐‘› −  โˆ†๐‘ก

 

๐‘ข๐‘›(๐‘ข๐‘› − ๐‘ข๐‘›   ) + โˆ†๐‘ก

 

(๐‘ข๐‘› − 2๐‘ข๐‘› + ๐‘ข๐‘›  )

 

 ๐‘— ๐‘—

 

2โˆ†๐‘ฅ  ๐‘— 

 

๐‘—+1

 

 

 

๐‘—−1

 

 

 

 (โˆ†๐‘ฅ)2

 

 ๐‘—+1

 

 ๐‘— ๐‘—−1

 

 Note, here the Fourier number is

 

โˆ†๐‘ก

 

๐น๐‘œ =

 

(โˆ†๐‘ฅ)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

 

๐œ•๐‘ข ๐œ•๐ธ ๐œ•2๐‘ข

 

๐œ•๐‘ก + ๐œ•๐‘ฅ = ๐œ•๐‘ฅ2

 

 ๐ธ =

 

 1 ๐‘ข2

 

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

 

Predictor: 

 

๐‘ข๐‘ = ๐‘ข๐‘› − โˆ†๐‘ก (๐ธ๐‘›   − ๐ธ๐‘›) + โˆ†๐‘ก

 

(๐‘ข๐‘› − 2๐‘ข๐‘› + ๐‘ข๐‘›  )

 

 ๐‘— ๐‘—

 

 โˆ†๐‘ฅ

 

 ๐‘—+๐‘– ๐‘—

 

 (โˆ†๐‘ฅ)2

 

 ๐‘—+1

 

 ๐‘— ๐‘—−1

 

 Corrector:

 

 ๐‘ข๐‘ = ๐‘ข๐‘ − โˆ†๐‘ก (๐ธ๐‘   − ๐ธ๐‘) + โˆ†๐‘ก

 

 (๐‘ข๐‘ − 2๐‘ข๐‘ + ๐‘ข๐‘  )

 

 ๐‘— ๐‘—

 

 โˆ†๐‘ฅ ๐‘—+๐‘– ๐‘—

 

 (โˆ†๐‘ฅ)2

 

 ๐‘—+1

 

 ๐‘— ๐‘—−1

 

 Then

 

 ๐‘ข๐‘›+1 = 0.5(๐‘ข๐‘ + ๐‘ข๐‘)

 

 ๐‘— ๐‘— ๐‘—

 

Note the above scheme is stable according to

 

1

 

โˆ†๐‘ก ≤ 1 2

 

โˆ†๐‘ฅ + (โˆ†๐‘ฅ)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.

(5/5)
Attachments:

Related Questions

. The fundamental operations of create, read, update, and delete (CRUD) in either Python or Java

CS 340 Milestone One Guidelines and Rubric  Overview: For this assignment, you will implement the fundamental operations of create, read, update,

. Develop a program to emulate a purchase transaction at a retail store. Thisย  program will have two classes, a LineItem class and a Transaction class

Retail Transaction Programming Project  Project Requirements:  Develop a program to emulate a purchase transaction at a retail store. This

. The following program contains five errors. Identify the errors and fix them

7COM1028   Secure Systems Programming   Referral Coursework: Secure

. Accepts the following from a user: Item Name Item Quantity Item Price Allows the user to create a file to store the sales receipt contents

Create a GUI program that:Accepts the following from a user:Item NameItem QuantityItem PriceAllows the user to create a file to store the sales receip

. The final project will encompass developing a web service using a software stack and implementing an industry-standard interface. Regardless of whether you choose to pursue application development goals as a pure developer or as a software engineer

CS 340 Final Project Guidelines and Rubric  Overview The final project will encompass developing a web service using a software stack and impleme

Get Free Quote!

291 Experts Online