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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

{[ snackBarMessage ]}

### Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online