Abstract: The development of practical numerical methods for simulation, control and optimization of partial differential equations leads to problems of convergence, accuracy
(in time and space) and efficiency. Verification of a computational algorithm consist
in part of establishing a convergence theory for the discretized equations. Convergence
analysis for simulation of linear systems is often based on some form of the Lax Equivalence Theorem, or its functional analysis version, the Trotter-Kato Theorem. Moreover,
it is well known that the long time behavior of a dynamical system may not be captured even by convergent approximating methods and additional requirements must
be placed on numerical schemes to ensure the discretized equations capture the correct asymptotic behavior. These issues are amplified when the approximate dynamical
systems are to be used for optimization or control.
Convergence analysis for nonlinear systems is much more complex and sensitivity to
small parameters can greatly impact both finite time accuracy and long term dynamic
predictions. Moreover, all modern computations are conducted on finite precision
digital computers and even theoretically convergent schemes can produce incorrect
dynamics. In this talk we provide some motivating applications involving modeling and
control of industrial heat exchangers, discuss how the intended use of the numerical
model can impact the choice of numerical approximation schemes and provide examples
to illustrate possible pitfalls that can occur with finite precision computations.
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