Pseudospectra of a Spectral Approximation of an Orr-Sommerfeld Operator
This figure illustrates pseudospectra of an Orr-Sommerfeld operator generated from a spectral method. This discretization results in a matrix of dimension N=99. There are numerous spurious eigenvalues beyond the plotted axis, so the matrix is projected onto an appropriate invariant subspace to accelerate the computation. Pseudospectra of Orr-Sommerfeld operators were first investigated by Reddy, Schmidt, and Henningson [RSH93].
Use the following MATLAB code compute a similar image using EigTool.
R = 5772; N = 100; [D,x] = cheb(N); % cheb.m from Trefethen's "Spectral Methods in MATLAB" D2 = D^2; D2 = D2(2:N,2:N); S = diag([0; 1 ./(1-x(2:N).^2); 0]); D4 = (diag(1-x.^2)*D^4 - 8*diag(x)*D^3 - 12*D^2)*S; D4 = D4(2:N,2:N); I = eye(N-1); A = (D4-2*D2+I)/R - 2i*I - 1i*diag(1-x(2:N).^2)*(D2-I); B = D2-I; C = B\A; opts.npts=50; opts.ax = [-1 .2 -1.05 0]; opts.levels = -10:.5:-2; opts.proj_lev = 1; eigtool(C,opts)Download this code: orrsomm.m.
Download cheb.m from L. N. Trefethen, Spectral Methods in MATLAB, SIAM, Philadelphia, 2000.
See also J. A. C. Weideman and S. C. Reddy, "A MATLAB differentiation matrix suite", ACM Transactions on Math. Software, to appear, and program