Page 188 - Process Modelling and Simulation With Finite Element Methods

Simulation und Nonlinear Dynamics 175

So is there any point in using PDE based models to describe complex
systems for which the complexity is practically indescribable? Of course, we
can derive or pose PDEs for the dynamics of the statistics (traditional turbulence
modeling) or to collate the statistics of the dynamics. In meteorology, the latter
is termed ensemble forecasting, and it is an attempt to quantify likely behavior of
emergent properties, rather than to average out uncertainty.

Stability and Eigenanalysis: Time Asymptotic Behavior
Key to the evolution of nonlinear systems is the notion of stability. A state of a
system, uo(t), is said to be stable if small perturbations, 6, do not displace the
new state of the system, u(t), very far from the original state. The concepts of a
‘state’, how you measure ‘small’ and ‘how far’ two states are separated need to
be precisely defined for stability (and therefore instability) to be a useful
concept.
The operational definition of a state u(t) is simply to list all of the degrees of
freedom necessary to uniquely define a recurring pattern in the system. For a
FEM model, this means giving the time dependence of a solution which is
typically either stationary or periodic. The exception is that a chaotic attractor is
also a ‘state’ of a dynamical system, deterministically known as a solution
trajectory u(t) from an initial state, but not uniquely defined as the attractor is an
‘asymptotic state’ - many initial conditions are attracted after a long time to this
state. In fact, the states of FEM models are easier to describe than for the
underlying pde system, which is inherently infinite dimensional. Once the trial
functions and finite elements are chosen, a FEM model is finite dimensional and
the degrees of freedom necessary to define a state is just the space of all possible
solution vectors.
In terms of FEM models, it is also straightforward to describe the stability of
a solution trajectory u(t). Consider the FEM operator that maps the solution at
time t to the solution at time t+At:

N{U (t)} = u (t + ~t) (5.1)
Conventionally, for small time steps, this operator can be linearized, so that
when applied to the perturbed system, we can compute
N{u(~) + 6} = u (t + ~t)+~6 (5.2)

where L is the Jacobian of N

aNi
L.. =- (4 (5.3)
‘1 auj