Spectral Methods in CFD

Proving Exponential Convergence with Chebyshev Methods

1. Prove that Chebyshev methods have exponential convergence rate for smooth functions (you can follow the lecture notes at the end of Lecture 18, but please, make sure that you understand all the steps).

2. Here you will work with the following functions:
(a) f (x) = x e−x, −1 ≤ x ≤ 1.
(b) f (x) = sin (π x), −1 ≤ x ≤ 1.
(c) f (x) = { 1 + x −1 ≤ x < 0
1 − x , 0 < x ≤ 1 .
(d) f (x) = { 1 + x −1 ≤ x < 0
−1 + x , 0 < x ≤ 1 .
In this problem, I encourage you to use your best judgement to show your results (plots are preferred), but all results need to be demonstrated.
(a) Show analytically that with Chebyshev discretization, the clustering of grid points near the bound- aries follows the rule ? x ∼ C/N 2, where C is some constant.
(b) Write a program that computes the Chebyshev transform, and test your program by transforming 1, x3, and x6. Compare the coefficients with exact coefficients. Also, write the program that computes inverse Chebyshev transform and test it on the same functions. Do you recover the original functions?
(c) Use your program to compute and plot the Chebyshev expansion coefficients for the four functions
(a) through (d). Use N = 4, 8 and 16. Comment on decay rate. Perform inverse Chebyshev transform. Do you recover the original functions?
(d) Write a program to calculate the first and second derivative using Chebyshev transform with recursive method covered in class. Test your program by differentiating polynomials x, x3, and x6. Use your program to differentiate the four functions (a) through (d). Take N = 4, 8, 16, 32 and compare to the exact derivative. Plot your derivatives. Look at convergence rate (by plotting L2 error for the derivatives versus N ). Discuss your results.
(e) Extra credit. Compute the first and second derivatives of all the functions (both the test functions x, x3, and x6 and the functions stated in the beginning of the problem) using a closed- form expression presented in Problem 3 (for the first derivative) and in the lecture notes (for the second derivative). Make sure both differentiation methods give you exactly the same answer.

3. A closed-form expression for the Chebyshev derivative coefficients is given by the following formula
bm = 2 cm N∑ p=m+1 p+m=odd p ap,
where ap are the Chebyshev coefficients of some smooth function f (x) and bm are the Chebyshev coefficients of f ′(x). Use mathematical induction to show that. Hint: start from the end and go backwards.

4. Solve the differential equation
∂ u
∂ t + 3 ∂ u
∂ x = 0
for u(x, t) on the domain −1 ≤ x ≤ 1 subject.

(a) Initial and boundary conditions
u(x, 0) = 0, u(−1, t) = sin πt.
The exact solution is
u = { 0 x ≥ −1 + 3 t sin π(t − x+1 3 ) −1 ≤ x ≤ −1 + 3 t.
(b) Initial and boundary conditions
u(x, 0) = sin π x 2 , u(−1, t) = sin πt − 1.
For this case, you have to find an exact solution yourself.

-Use Chebyshev collocation, Chebyshev tau and a finite-difference method of your choice. Use the same time stepping method for all the three schemes. Plot the solution at several t, try 1/3, 1/2, 2/3, and 7/8. Plot the L2 error at t = 7/8 versus N for the three methods. Compare the accuracy. When you plot the solutions and the error, plot the results from all the three methods on the same plot (but separately for each problem), for better comparison. What time step did you use? What was the maximum time step you could use? Don’t use Matlab built-in functions for solution of ODEs, write your own routines for time-stepping. All questions must be answered in order to receive full credit. <span dir="ltr" style="left: