University of Leeds
Workshop on Geometric Integration, 4th May, 1999
Solving linear differential equations in Lie groups
Abstract
The subject matter of this talk is the numerical solution of
differential equations that evolve on Lie groups. Such equations are
important in many applications, when it is important to respect
invariants and symmetries.
We will demonstrate that the solution can be pushed to the underlying
Lie algebra and expanded there in Magnus series. The structure of the
series and their construction use graph theory. The discretization of
the expansion requires a range of techniques in multivariate
integration, combinatorics and the theory of free Lie algebras.
The outcome is a technique that allows for a high-precision
integration of complicated differential equations at an affordable
cost and that has been already applied, e.g. to the determination of
Sturm--Liouville spectra and to integration in homogeneous spaces.
Incorporating symmetry and the maximum principle into a numerical
scheme
Abstract
In the theory of partial differential equations two very
important features governing the qualitative behaviour of
their solutions are the underlying Lie symmetries and
the maximum principle. The first because it allows a reduction
of the system to an ordinary differential equation with
resulting self-similar solutions. The second because it
gives an ordering on the solutions which often impleis that the
self-similar solutions are global attractors.
If a numerical method is to accurately reproduce the qualitative
behaviour of a partial differential equation it is desirable
that it should be both invariant under the action of the symmetry
group and also posess a maximum principle. In practice it is
hard to achieve both of these two desirable qualities simultaneously.
However in certain special case it can be done. We will describe
a discretisation of the porous medium equation that not only
has symmetry invariance and a maximum principle but also retains
certain conservation laws. As a result is has excellent global
qualitative features.
This is joint work with Matthew Piggott (Bath)
Orthogonal Integration and the Computation of Lyapunov Exponents
Abstract
The problem of computing approximate
Lyapunov exponents has been addressed in various
application areas, especially in the physics community,
but it has received relatively little attention from
numerical analysts. Hence the convergence theory is incomplete.
In this talk we focus on the discrete QR iteration,
which has two main elements:
- the solution of an ODE with a built-in orthogonality structure, and
- the approximation of an infinite-time integral.
A well-designed algorithm must balance the error contributions from
the numerical ODE solver and the quadrature process.
To get some insight into how this balancing act may take place, we
present a convergence analyse for a simple class of test problems.
The analysis raises an intesting subproblem in numerical linear algebra.
Multi-Symplectic Integrators: Numerical Methods That Make Waves
Abstract
A wide range of conservative PDEs can be formulated as a
multi-symplectic Hamiltonian PDE. Contrary to the "classical"
approach to Hamiltonian PDEs as an evolutionary system on an
appropriate function space, the multi-symplectic formulation
treats both time and space as local quantities leading to
local conservation laws for symplecticity, energy, and
momentum. In my talk I will present numerical schemes that
lead to a discrete conservation of symplecticity and
excellent conservation of energy and momentum. This work
can be seen as a natural extension of symplectic integration
for classical mechanics to Hamiltonian PDEs.
Canonical integrators and the search for good actions
Abstract
Modern numerical Lie group integrators for differential equations are
formulated in an abstract language, which may be interpreted in different
concrete settings. We will give an introduction to various views of these
methods. Whereas classical integration schemes are always based on
stepping forwards by translations on R^n, the new scemes allow a wide
range of different actions for advancing the solution. The choice of
actions is crucial for the properties of the numerical solution, and the
problem of choosing a good action can be compared to the problem of finding
a good preconditioner in an iterative solver for a linear system. In both
cases, one wants approximations to the original system that are easy to
solve and that captures some essential properties of the system we want to
integrate. We will through various examples show the importance of finding
good actions.
Contact Carsten Knudsen (email: carsten@amsta.leeds.ac.uk ) or Vadim Kuznetsov (email:
vadim@amsta.leeds.ac.uk )
for a TEX file of the programme and further details.
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