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mid-term-exam-sol

mid-term-exam-sol - ME 608 Numerical Methods in Heat Mass...

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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 ∂φ t + x ( u φ ) = 0 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
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(a) Using the implicit CDS, the discrete equation for point P may be written as φ n + 1 P - φ n P Δ t + u φ n + 1 E - φ n + 1 W 2 Δ x = 0 (b) Our Taylor series expansion is centered at time level ( n + 1 ) at point P . Thus φ n P = φ n + 1 P - Δ t φ n + 1 t , P + Δ t 2 2! φ n + 1 tt , P + ... φ n + 1 E = φ n + 1 P + Δ x φ n + 1 x , P + Δ x 2 2! φ n + 1 xx , P + Δ x 3 3! φ n + 1 xxx , P + ... φ n + 1 W = φ n + 1 P - Δ x φ n + 1 x , P + Δ x 2 2! φ n + 1 xx , P - Δ x 3 3! φ n + 1 xxx , P + ... Manipulating the above equations, we may write φ n + 1 P - φ n P Δ t = φ n + 1 t , P - Δ t 2! φ n + 1 tt , P + ... u φ E - φ P 2 Δ x n + 1 = u φ n + 1 x , P + u Δ x 2 3! φ n + 1 xxx , P + ... Substituting into the discrete equation for point P , we obtain φ t + u φ x = φ tt Δ t 2! - u φ xxx Δ x 2 3! + ... We want to convert φ tt to φ 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. φ tt + u φ xt = 0 u φ xt + u 2 φ xx = 0 Subtracting the second equation from the first yields φ tt = u 2 φ xx Combining this relationship with our model equation, we obtain φ t + u φ x = u 2 Δ t 2! φ xx - u φ xxx Δ x 2 3! + ... The temporal truncation error is O ( Δ t ) . The spatial truncation error is O ( Δ x 2 ) . (c) The scheme is dissipative because the leading order truncation term is a diffusion term with an artificial diffusion coefficient of u 2 Δ t 2 .
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