Examples from engineering

See also Examples of pseudospectra, and the

Below are three examples taken from engineering problems available at Matrix Market. The examples are:

• The Brusselator wave model from chemical engineering (dimension 5000)
• The diffusion model study for crystal growth simulation (dimension 10,000)
• Transient stability analysis of a Navier-Stokes Solver (dimension 23,560)

Brusselator wave model

This matrix is the Jacobian from a model of the concentration waves for reaction and transport interaction of chemical solutions in a tubular reactor. Stable periodic solutions exist for a parameter when the rightmost eigenvalues of the Jacobian are purely imaginary. For a matrix of dimension 5000, using a subspace of dimension 100 and asking eigs for 20 eigenvalues, we obtain the eigenvalue estimates and approximate pseudospectra shown above. Although the matrix is evidently slightly more non-normal at the right hand side, the non-normality is mild, and the conclusion from this computation is that the Ritz values returned by eigs are accurate and that the computation has been successful.

Diffusion model study

The matrix here arises in a stability analysis of a crystal growth problem. The eigenvalues of interest are the ones with largest real part. The fact that we can see the 10^-13 pseudospectrum (when the axis scale is O(1)) indicates that this matrix is significantly non-normal, and although the matrix is too large for us to be able to compute its exact pseudospectra for comparison, this is certainly a case where the non-normality could be important, making all but the rightmost few eigenvalues non-physical. The eigs parameters we used in this case were p=80 (subspace size) and k=30 (number of requested eigenvalues), and the computation took about one hour on our Sun Ultra 5 workstation.

As far as correlation to the true pseudospectra of this matrix, we would expect that the rightmost portion should be fairly accurate, where there is a good deal of Ritz data and relatively little non-normality, losing accuracy at the left where the effect of the remaining eigenvalues of the matrix unknown to the approximation begins to become important, as well as the increased non-normality making convergence more difficult.

Transient stability analysis

The third example (above) shows the pseudospectra of the matrix created by performing transient stability analysis of a Navier-Stokes solver. In this case the matrix appears fairly close to normal, and the picture gives every reason to believe that the eigenvalues have physical meaning. Using p=80 and k=30, eigs took about 9 hours to converge to 18 eigenvalues of the matrix, whilst we were able to plot the pseudospectra in about 3 minutes (even on the fine grid used here).

Pseudospectra GUI home page.