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Runge-Kutta methods and differential-algebraic equations
Laurent Jay
University of Iowa
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In this series of lectures I will discuss in detail the numerical
approximation of solutions to a broad class of differential-algebraic
equations (DAEs) by a family of Runge-Kutta methods.
DAEs represent differential equations on manifolds and they generally
consist of a system of differential equations coupled with
linear/nonlinear equations representing constraints,
conservations laws, etc. For example the model equations of
systems in mechanics do not generally take the form of ordinary
differential
equations, but of DAEs (of mixed "index 3 and 2") due to the presence
of
geometrical and kinematic constraints. The family of Runge-Kutta methods
that
will be presented has the advantages of not only preserving the manifold
of
constraints itself, but also geometric properties in phase space of
certain
classes of DAEs, such as: the Poisson/symplectic structure
of the flow of Hamiltonian systems; the variational nature of trajectories
of
Lagrangian systems; reversibility properties of the flow for
DAEs with symmetries. Those are invariants of a different nature than
simple linear/nonlinear equations. Such invariants cannot be directly
enforced,
but only indirectly through the numerical integrator properties. Methods
preserving intrinsic properties of differential equations have generally
superior
qualitative and quantitative behaviors. This corroborates an illuminating
and general statement by Richard Wesley Hamming dating back to 1973:
"... an algorithm which transforms properly with respect to a class of
transformations is more basic than one that does not. In a sense the
invariant algorithm attacks the problem and not the particular
representation used..."
Link
to some related publications.
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