10.1

In our stability analysis of Program 6, we "froze the coefficients" and assumed that the largest eigenvalue of the discretization of u _t + c( x) u _x = 0 would be about 6/5 times that of the discretization of u _t + u _x. Perform a numerical study to investigate how true this is. (You may wish to work with a matrix formulation of the algorithm, as in Exercise 3.7.) Produce a plot of the ratio of the actual and estimated eigenvalues for N = 20, 40, 60, , 200. For N = 128, how does the true eigenvalue compare with the frozen-coefficient prediction? What stability restriction does the true eigenvalue suggest? Does this match the empirically observed stability restriction? Does the corresponding eigenvector look like the unstable mode visible in Figure 10.1?

10.2

Rerun Program 27 with the time step increased from 0.4 N ^?2 to 0.45 N ^?2. Comment on the resulting plot. Can you explain this effect with reference to stability regions?

10.3

Consider the first-order linear initial boundary value problem

with initial data u( x, 0) = exp( ?60( x ? 1/2) ²). Write a program to solve this problem by a matrix-based Chebyshev spectral discretization in x coupled with the third-order Adams{Bashforth formula in t, for which the formula is . Initial values can be supplied from the exact solution.