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# pb4_10 - MPO 662 Problem Set 4 1 Consistency and stability...

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MPO 662 – Problem Set 4 1 Consistency and stability You have reverse-engineered a computer program and deciphered that it updates the variable u according to the following rule: u n +1 j - u n j Δ t + c 3 u n j - 4 u n j - 1 + u n j - 2 x = 0 (1) Determine the continuous PDE this finite difference equation is trying to approximate. Determine the truncation error for this scheme and derive the leading terms of the modified equations. Determine the stability characteristics of this scheme. With the time-derivative kept in continuous form, study the dispersion characteristics of the spatial discretization. 2 Heat equation The concepts of stability, consistency and convergence extend readily to parabolic equations. Here we will study several instances of such equations. The 1D heat equation, u t = νu xx , is discretized using a θ -scheme in time according to: u n +1 j - u n j Δ t = νθ u n +1 j +1 - 2 u n +1 j + u n +1 j - 1 Δ x 2 + ν (1 - θ ) u n j +1 - 2 u n j + u n j - 1 Δ x 2 (2) 1. Derive the Taylor series of this scheme around the space-time point ( x n , t n + 1 2 ) and show that the scheme is second-order when θ = 1 / 2, and first-order otherwise. Hint: Start with the spatial expansion and then the temporal one. 2. Derive the amplification factor as a function of θ and deduce the stability conditions for the cases θ = 0 , 1 / 2 and 1. 3. Discuss the algorithmic and computational advantages or disadvantages of each choice of θ . 4. Plot the decay factor as a function of k Δ x/π for the three different values of θ and compare it to that of the analytical factor A a = e - αk 2 Δ x 2 where α = ν Δ t Δ x 2 . Try different values of α : 0.1 0.25, 0.5, 0.75, 1 and 10 (please use only those that are stable in the

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