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Unformatted text preview: ME 608 Numerical Methods in Heat, Mass, and Momentum Transfer Mid-Term 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!...
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This note was uploaded on 12/29/2011 for the course ME 608 taught by Professor Na during the Fall '10 term at Purdue University-West Lafayette.
- Fall '10