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HW4[1]

# HW4[1] - MAE 290B Winter 2010 HOMEWORK 3 Due Wed in class...

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MAE 290B, Winter 2010 HOMEWORK 3 Due Wed 02-24-2010 in class Provide source codes used to solve all problems PROBLEM 1 Consider the advection - diffusion equation t u + c∂ x u = ν∂ xx u, 1. Discretize this equation spatially using 2nd-order centered finite difference formulae for the convective terms and the diffusive terms. 2. Discretize the resulting set of ODEs temporally using a low-storage, mixed RK3- θ method. Express this discretization as a function of the two non-dimensional parate- mers CN = c Δ t/ Δ x (Courant Number) and V N = ν Δ t/ x ) 2 . 3. Apply Von-Neumann stability analysis to determine the stability of the resulting spatio- temporal scheme. For this purpose, (a) Find the amplification factor σ = u n +1 j /u n j as a function of CN , V N and k Δ x . (b) Plot the contour | σ | = 1 as a function of k Δ x and CN for V N = 0 , 5 , 10 , 20. Use this plot to discuss the stability of the scheme (c) Consider now the case V N = 0. Find the maximum value of CN for which the scheme is stable. Relate this result to the stability region of the RK3 scheme.

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