All cool, but let’s start with some simple and known algebra background of the modal eigenvalue kind:
What exactly do eigenvalues offer that makes them useful for so many problems? We believe there are three principal answers to this question, more than one of which may be important in a particular application.
Diagonalization and separation of variables
Diagonalization and separation of variables: use of the eigenfunctions as a basis. One thing eigenvalues may accomplish is the decoupling, of a problem involving vectors or functions into a collection of problems involving scalars, which may make subsequent computations easier. For example, in Fourier’s problem of heat conduction in a solid bar with zero temperature at both ends, the eigenmodes are sine waves that decay independently as a function of time. If an arbitrary initial temperature distribution is expanded as a sum of these sine waves, then the solution at a later time can be calculated by summing the components of the expansion.
Resonance: heightened response to selected inputs.
Diagonalization is an algorithmic idea; the other uses of eigenvalues are more physical. One is the analysis of the phenomenon of resonance, perhaps most familiar in the context of vibrating strings,
drums, and mechanical structures. Any visitor to science museums has seen demonstrations showing that certain systems respond preferentially to vibrations at special frequencies. These frequencies are the eigenvalues of the linear or linearized operator that governs the system in question, and the form of the response is associated with the corresponding eigenfunctions. Examples of resonance are familiar: One thinks of soldiers breaking step as they cross bridges; of the less fortunate Tacoma Narrows Bridge in the 1940s, whose collapse was initiated by a wind-induced flow oscillation too close to a structural eigenfrequency; of buildings and their response to the vibrations of earthquakes. Essentially there the wide range of complexity in applications of eigenvalue ideas, for the radio problem is straightforward and almost perfectly linear, whereas the ear is a complicated nonlinear system, not yet fully understood, for which eigenmodes are only a crude first step.
Asymptotics and stability: dominant response to general inputs.
A related application of eigenvalues is to questions of the form, What will happen as me elapses (or in the extreme, t → ∞) to a system that has experienced some more or less random disturbance?
Fourier’s heat problem is an awesome example: Whatever the shape of the initial temperature distribution, the higher sine waves decay faster than the lowest one, and therefore almost any initial
distribution will eventually come to look like the half-wavelength sine with zeros just at the two ends of the interval. Sometimes the crucial issue is a question of stability: Are there modes that grow rather than decay with time?
For example, in fluid mechanics a standard technique to determine whether small perturbations to a laminar flow will be amplified into large ones—which may then trigger the onset of turbulence—
is to calculate whether the eigenvalues of the system all lie in the left half of the complex plane. (This is of course orr-sommerfeld/Squire analysis).
We may think of fourth reason for the success of eigenvalues:
It could be such that it is equivalent to the condition that the eigenvector of A (if they even exist), are far from being orthogonal.
The other type on the spectrum are the Hermitian matrices with a complete set of orthogonal eigenvectors.
In the first case, for which the eigenvectors are far from normal, an eigenvalue analysis may certainly fail, and as it seems, even though many examples for eigenvalue analysis are a productive means for
investigation, it is our fluid mechanics where we tend this failure apparent in some of the problems (one references which is one of my favorite is Böberg & Brosa 1988; Reddy & Henningson 1993;Trefethen et al. 1993).
nonmodal instability of certain discretizations of time-dependent partial differential equations.
In this section we describe the phenomenon by presenting theorems to characterize discretizations that are free of this effect. Below is a numerical solution of a The standard technique for explaining the instability of finite difference formulas was developed by von Neumann and others and
described in a 1951 paper of O’Brien, Hyman, Kaplan. ‘Von Neumann analysis’ is another term for discrete Fourier analysis.
One begins by noting that if we ignore the complication of boundary conditions and imagine that the domain is unbounded, then any initial condition for the finite difference formula can be written as a superposition of waves.
For real wave numbers k and corresponding amplification factors (i.e., eigenvalues) λ = λ(k). If |λ| > 1 for some k, we have exponential growth and instability.
It should be very clear though:: Convergence of PDE discretizations depends on norm-boundedness of families of matrices.
It is explainable how von Neumann analysis fits into the general theory of Lax stability:
i.e. the famous Lax Equivalence Theorem, which states that if the discrete approximation is consistent, meaning that it approximates the right PDE as Δx → 0 and Δt → 0, then convergence ⇐⇒ stability.
Here ‘stability’ means that the solution operators are uniformly bounded as the time and space grid sizes approach zero.
A priori, the question of stability requires the analysis of families of matrices, and eigenvalue analysis alone could never give bounds on norms of powers of arbitrary families of matrices. In the special case of constant-coefficient problems on regular grids, however, the Fourier transform takes what would be families of matrices of unbounded dimensions in space into families of matrices of a fixed dimension, indexed over wave numbers. The transformation is unitary, and as a consequence, eigenvalue analysis of the resulting matrices is enough to ensure stability. For practical problems involving boundaries or variable coefficients, further theorems have been proved to show that von Neumann analysis still gives the correct results provided certain additional assumptions are satisfied
such as smoothness of coefficients.
On the other hand, there are some discretizations of PDEs that are fundamentally not translation-invariant. For these, von Neumann analysis is inapplicable, and instabilities may appear that are
nonmodal in nature.
Let’s have a look at an example:
The first figure reveal that if Δt = O(N −2) as N → ∞, then the maximal norm, though possibly large, is uniformly bounded for all N.
The discretization is Lax-stable, and the numerical solution will converge to the exact solution in the absence of rounding errors. If Δt = O(N −2) as N → ∞, on the other hand, there will be Lax instability
and no convergence. In particular, a choice such as Δt = 0.4N −1 will be catastrophic, even though the eigenvalues in that case remain inside the unit disk for all N. The second figure shows the pseudospectra of S for the particular choice N = 20 and Δt = 0.4N −1 = 8N −2 (thus S has dimension 60). Around most of the unit circle, the resolvent norm is of modest size, but in the region z ≈ −1 it takes values beyond 106, making it clear that there must be large transient growth. Since the boundary of the pseudospectrum crosses The main question that should be asked is if could one use a discretization of this kind for large-t simulations, since the instability is transient and dies away eventually? At a glance it might seem so, but the instability can only be expected to be transient for a purely constant-coefficient linear problem in the absence of rounding errors in other words As soon as variable coefficients or nonlinearities or other perturbations are introduced, the loss of convergence is likely to become global.
Conclusion of Part I
Fluid flows that are smooth at low speeds become unstable and then turbulent at higher speeds. This phenomenon has traditionally been investigated by linearizing the equations of flow and looking for unstable eigenvalues of the linearized problem, but the results agree poorly in many cases with experiments. Nevertheless, it has become clear that linear effects play a central role in hydrodynamic instability. A reconciliation of these findings with the traditional analysis can be obtained by considering the “pseudospectra” of the linearized problem, which reveal that small perturbations to the smooth flow in the form of streamwise vortices may be amplified by factors on the order of 10′ by a linear mechanism,
even though all the eigenmodes are stable. The same principles apply also to other problems in the mathematical sciences that involve non-orthogonal eigenfunctions.