Design and Implementation of Object-Oriented ... - Automatic Control

Chapter 2. Modeling Techniques
In the case of elliptic PDE, time derivatives do not occur and only space
coordinates are independent variables. They are a rare exception in dynamical
systems modeling. It should be pointed out again that the type
of the equation may change on Ω because Δ depends on x and t.
Numerical Methods for PDE The numerical solution of PDE is generally
a difficult problem. Solutions are always approximations to the true
mathematical solution of the equation and all the more to the real physics
that they represent. Solutions can be very sensitive to small changes in
parameters or boundary conditions. The situation is not hopeless, there
are several numerical methods which can solve technically relevant problems
reliably. There is active research in the area of solutions to PDE and
from the many methods available only the key features of a few of them
will be discussed.
A generalized concept that comprises the majority of numerical PDE
solution methods is the “Method of Lines”. The methods of lines converts
time dependent PDEs into ODEs (or DAEs) with respect to time
by choosing a fixed spatial discretization and usually also a fixed order
approximation to the spatial derivatives of u(x, t). The name “Method of
Lines” is based on the fact that many of these methods approximate the
solution of the equations on a fixed spatial grid of straight lines. Method
of lines as a group name also refers to finite element methods where the
spatial discretization is fixed, but the discretization uses triangles of different
sizes. Traditional PDE solvers also use a fixed time discretization,
but in recent years it has become common to use variable time step ODE
solvers for discretized PDE.
The method of lines can provide good numerical approximations to the
solution of PDE in a wide variety of applications. The family of methods
of lines comprises finite difference, finite volume, finite element and
weighted residual methods in which piecewise local or global approximation
functions in the space dimension are used to convert hyperbolic
and parabolic PDE problems into initial value ODE problems. The key
advantage of the “Method of Lines” is twofold:
• It is straightforward to combine subsystem models which are described
by DAEs or ODEs with PDEs because they are all converted
to the same mathematical form.
• ODE and DAE solvers can be employed for the solution. These
solvers have reached a higher degree of sophistication than PDE
solvers. Many programs are available which permit fast and accurate
solutions to large sets of DAE [Petzold, 1982].
The advantage of ODE/DAE solvers is that they use sophisticated algorithms
to adjust the time step size and the order of a method in order to
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