HW3 - hybrid RKW3-θ scheme to integrate this problem 2...

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MAE 290B, Winter 2010 HOMEWORK 3 Due Mon 02-01-2010 in class Provide source codes used to solve all problems PROBLEM 1 Discretize the real functions u ( x ) = cos(10 x ), v ( x ) = cos(8 x ), and w ( x ) = u ( x ) v ( x ) over the interval 0 x < 2 π , on the grid points x j = jL/N for j = 0 ...N - 1 where N = 64. Compute and plot the magnitude of the Fourier coefficients b u n , b v n , b w n . 1. Can you show that the nonlinear product w ( x ) = u ( x ) v ( x ) transfers energy towards lower and higher frequencies? 2. Would aliasing happen if N = 16? Explain why and compare to the case N = 64. PROBLEM 2 Consider a problem in which we expand u ( x ) as a Fourier series. Find the zero-padding dealiasing rules that should be employed to avoid aliasing in non-linear terms u 2 ( x ) and u 3 ( x ). PROBLEM 3 Consider Burgers equation t u + u∂ x u = ν∂ xx u, in the domain 0 x < 2 π , where ν = 10 - 3 and with initial condition u ( x, 0) = sin( x ). 1. Implement a pseudospectral Fourier method with 3/2 dealiasing and a low-storage
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Unformatted text preview: hybrid, RKW3-θ scheme to integrate this problem. 2. Estimate the maximum value of Δ t = Δ t max that can be used to integrate this problem. 3. Solve the problem for 0 ≤ t ≤ 1 using N = 16 , 64 , 128 , 2048 grid points in x and Δ t = 0 . 2Δ t max . 4. Compare the solutions obtained in the previous step at t = 0 . 25 , . 5 , . 75 , 1 in terms of u ( x ) and | b u n | . Explain their time evolutions and their differences. 5. Is there Gibbs error in the previous solutions? Why? 6. Include a 6 th-order hyperviscosity regularization term ±∂ 6 x u in the equation with ± = 10-3 ν and repeat steps 2-5. Discuss the effect of the regularization term. 7. Remove the non-linear term u∂ x u and repeat steps 2-5. Discuss the effect of removing this term in the solution....
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HW3 - hybrid RKW3-θ scheme to integrate this problem 2...

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