This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: ME 608 Numerical Methods in Heat, Mass, and Momentum Transfer MidTerm Exam Solution Date: March 9, 2011 6:00 8:00 PM Instructor: J. Murthy Open Book, Open Notes Total: 100 points 1. Consider the 1D linear wave equation t + x ( u ) = and the stencil of control volumes shown in Fig. 1. Assume u > 0. Figure 1: Computational Domain for Problem 1 (a) Derive the discrete equation corresponding to the implicit central difference scheme. (b) Derive the model equation for the scheme. What is the order of spatial and temporal truncation error? Hint: Be sure that all your derivatives are written at the same time level, and are all located at the same grid point . (c) Using the model equation as a basis, discuss whether the scheme is dispersive or dissipative. (d) Do you need to stabilize the scheme? Justify your answer using the model equation as a basis. 1 (a) Using the implicit CDS, the discrete equation for point P may be written as n + 1 P n P t + u n + 1 E n + 1 W 2 x = (b) Our Taylor series expansion is centered at time level ( n + 1 ) at point P . Thus n P = n + 1 P t n + 1 t , P + t 2 2! n + 1 tt , P + ... n + 1 E = n + 1 P + x n + 1 x , P + x 2 2! n + 1 xx , P + x 3 3! n + 1 xxx , P + ... n + 1 W = n + 1 P x n + 1 x , P + x 2 2! n + 1 xx , P x 3 3! n + 1 xxx , P + ... Manipulating the above equations, we may write n + 1 P n P t ! = n + 1 t , P t 2! n + 1 tt , P + ... u E P 2 x n + 1 = u n + 1 x , P + u x 2 3! n + 1 xxx , P + ... Substituting into the discrete equation for point P , we obtain t + u x = tt t 2! u xxx x 2 3! + ... We want to convert tt to xx . We can do this by differentiating the pde wrt t to obtain the first equation below. We also differntiate the pde wrt x and multiply by u to obtain the second equation below. tt + u xt = u xt + u 2 xx = Subtracting the second equation from the first yields tt = u 2 xx Combining this relationship with our model equation, we obtain t + u x = u 2 t 2!...
View
Full
Document
This note was uploaded on 12/29/2011 for the course ME 608 taught by Professor Na during the Fall '10 term at Purdue UniversityWest Lafayette.
 Fall '10
 NA

Click to edit the document details