For a proper authoritative definition of ``well posed'' in the field of finite-difference schemes, see, e.g., [484]. The definition we will use here is less general in that it excludes amplitude growth from initial conditions which is faster than polynomial in time.
We will say that an initial-value problem is well posed if the linear system defined by the PDE, together with any bounded initial conditions is marginally stable.
As discussed in [452], a system is defined to be stable when its response to bounded initial conditions approaches zero as time goes to infinity. If the response fails to approach zero but does not exponentially grow over time (the lossless case), it is called marginally stable.
In the literature on finite-difference schemes, lossless systems are classified as stable [484]. However, in this book series, lossless systems are not considered stable, but only marginally stable.
When marginally stable systems are allowed, it is necessary to accommodate polynomial growth with respect to time. As is well known in linear systems theory, repeated poles can yield polynomial growth [452]. A very simple example is the ordinary differential equation (ODE)
which, given the initial condition
for any constant
When all poles of the system are strictly in the left-half of the
Laplace-transform
plane, the system is stable, even when
the poles are repeated. This is because exponentials are faster than
polynomials, so that any amount of exponential decay will eventually
overtake polynomial growth and drag it to zero in the limit.
Marginally stable systems arise often in computational physical
modeling. In particular, the ideal string is only marginally stable,
since it is lossless. Even a simple unaccelerated mass, sliding on a
frictionless surface, is described by a marginally stable PDE when the
position of the mass is used as a state variable (see
§7.1.2). Given any nonzero initial velocity, the position
of the mass approaches either
or
infinity, exactly as in the
example above. To avoid unbounded growth in practical
systems, it is often preferable to avoid the use of displacement as a
state variable. For ideal strings and freely sliding masses, force
and velocity are usually good choices.
It should perhaps be emphasized that the term ``well posed'' normally allows for more general energy growth at a rate which can be bounded over all initial conditions [484]. In this book, however, the ``marginally stable'' case (at most polynomial growth) is what we need. The reason is simply that we wish to excluded unstable PDEs as a modeling target. Note, however, that unstable systems can be used profitable over carefully limited time durations (see §9.7.2 for an example).
In the ideal vibrating string, energy is conserved. Therefore, it is a marginally stable system. To show mathematically that the PDE Eq.(D.2) is marginally stable, we may show that
for some constants
for all
Note that solutions on the ideal string are not bounded, since, for
example, an infinitely long string (non-terminated) can be initialized
with a constant positive velocity everywhere along its length. This
corresponds physically to a nonzero transverse momentum, which is
conserved. Therefore, the string will depart in the positive
direction, with an average displacement that grows linearly with
.
The well-posedness of a class of damped PDEs used in string modeling is analyzed in §D.2.2.