Homework 5

Submit via the Canvas page by 11:00pm on Friday December 7, 2015. Homework can be submitted up to 24 hours late with a 10% reduction in points possible. If special circumstances warrant turning in assignments late, please make arrangements in advance.

Problem 1

The bottom figure on page 172 of ATAP shows the approximate solution to the Airy equation (21.4) computed using a Chebyshev spectral method. The resulting Chebfun has length \(n=170\).

Write a program that computes the approximate solution for values of n = 10:10:200 and produces a semilog plot of the error vs. \(n\).

Note the following:

• Set up and solve an \(n\times n\) linear system for each value of \(n\). You can get the matrix by evaluating matrix(L,n-2) where L is the chebop defined on page 172, and then you will have to define the right hand side of the system appropriately to impose the desired boundary conditions. Examine such a matrix for small n to figure out how the boundary condition should be set.
• For the exact solution the matlab function airy can be used, but you will have to rescale the value returned by this function so that \(\hat p(x)\) at the left end point has the value 1, since this is the boundary condition imposed on the numerical solution.
• To measure the error, take the vector \(u_n\) obtained with \(n\) points and use Chebfun to generate the interpolating polynomial \(p_n(x)\) over the interval. Also define \(\hat p(x)\) to be the Chebfun approximation of the true solution (allowing it to use as many points as needed to get machine precision). Then define the error as \(E_n = \|p_n-\hat p\|_\infty\) (which can be computed with norm(pn-phat,inf) if these are defined properly as Chebfuns).

Also plot the computed and true solutions together on the same plot for selected values of \(n = 100, 110, 120\).

Problem 2

To see how convergence of the spectral method compares to convergence of the interpolation problem of approximating the Airy function, write a program that uses Chebfun to compute the polynomial \(p_n(x)\) that interpolates the Airy function at \(n\) Chebyshev points and plot the error \(E_n = \|p_n-\hat p\|_\infty\) vs. \(n\) on the same semilog plot as the results from Problem 1. Again do this on the interval \([-30,30]\) and scale the function you are interpolating to have \(\hat p(-30) = 1\) so that \(\hat p\) is the same as before.

Problem 3

The notebook “Spectral_Advection.ipynb” found in Sample codes and the codes directory of the class GitHub repository illustrates using the Chebyshev spectral method to solve the advection equation \(u_t = u_x\) using the Forward Euler method for time stepping.

See also the notebook “Dx_Pseudospectra.ipynb” for more discussion of spectra and pseudospectra of this operator both with and without boundary conditions.

Write a code for this same problem but using 4th order Runge-Kutta method for time stepping and investigate what time step restriction must be imposed to obtain a stable approximation, by plotting the eigenvalues of the spectral differentiation matrix being used and taking into account the stability region of the method (which can be plotted by adapting the code in Program 25 of SMM). Check also that the pseudospectral lie within \({\cal O}(\epsilon)\) of the stability region as \(\epsilon \rightarrow 0\).

Confirm this analysis of the allowable time step by running your code for various values of \(\Delta t\) to see when it is stable.

The Runge-Kutta method for \(u'(t) = f(u)\) is:

\(k_1 = f(U^n)\)

\(k_2 = f(U^n + 0.5 \Delta t k_1)\)

\(k_3 = f(U^n + 0.5 \Delta t k_2)\)

\(k_4 = f(U^n + \Delta t k_3)\)

\(U^{n+1} = U^n + \frac{\Delta t}{6} (k_1 + 2k_2 + 2k_3 + k_4)\)

For the problem \(u_t = u_x\) with boundary condition \(u(1,t) = \sin(5(1+t))\) with exact solution \(u(x,t) = \sin(5(x+t))\), explore how the error at time \(t=1.0\) behaves both as the number of grid points \(N\) is varied and as the time step \(\Delta t\) is varied.