Transient Growth

Nick Trefethen, July 2011

(Chebfun example linalg/TransientGrowth.m)

If A is a matrix whose eigenvalues are in the open left half of the complex plane, then the corresponding dynamical system defined by the equation du/dt = Au is asymptotically stable, with all solutions decaying to zero as t->infinity. Since the solution is u(t) = exp(tA)*u(0), another way to say this is that the quantities norm(expm(tA)) decay to zero as t->infty.

Along the way, however, there may be transient growth, and this is important for example in some problems in fluid mechanics. A recent paper by Whidborne and Amar  considers the following matrix taken from an earlier paper by Plitschke and Wirth:

tic
A = [-1 0 0 0 0 0 -625; 0 -1 -30 400 0 0 250; -2 0 -1 0 0 0 30;
5 -1 5 -1 0 0 200; 11 1 25 -10 -1 1 -200;
200 0 0 -150 -100 -1 -1000; 1 0 0 0 0 0 -1]
A =
Columns 1 through 6
-1           0           0           0           0           0
0          -1         -30         400           0           0
-2           0          -1           0           0           0
5          -1           5          -1           0           0
11           1          25         -10          -1           1
200           0           0        -150        -100          -1
1           0           0           0           0           0
Column 7
-625
250
30
200
-200
-1000
-1

Here (adapted from linalg/NonnormalQuiz.m) is a code to compute and plot norm(expm(tA)) as a function of t:

e = chebfun(@(t) norm(expm(t*A)),[0 2.5],...
'vectorize','splitting','on','eps',1e-14);
LW = 'linewidth'; FS = 'fontsize'; plot(e,'b',LW,2)
xlabel('t',FS,14), ylabel('||e^{tA}||',FS,14)
title('amplitude',FS,16) Actually Whidborne and Amar plot the square of this function. The following figure matches their Figure 1.

e2 = e.^2;
plot(e2,'b',LW,2)
xlabel('t',FS,14), ylabel('||e^{tA}||^2',FS,14)
title('energy',FS,16) They are interested in calculating the maximum energy:

fprintf('Maximum energy = %15.8f\n',max(e2))
Maximum energy = 358147.98785179

Here's the time for this Example:

toc
Elapsed time is 9.014774 seconds.

References:

 L. N. Trefethen and M. Embree, Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators, Princeton U. Press, 2005.

 J. F. Whidborne and N. Amar, Computing the maximum transient energy growth, BIT Numerical Mathematics 51 (2011), 447-457.