MaGIC 2005
Ustaoset, Norway
14-18 February

TJB Lecture Slides

Numerics of ODEs on exterior algebra spaces
Given a vector space V of dimension n, there are a number of other vector spaces that can be built on it: the dual space, the spaces of k-vectors, and k-forms, for k=0,...,n. Given a linear ODE on V it is often of interest to numerically integrate the induced systems on exterior algebra spaces. Such systems arise in the linearization of nonlinear ODEs about trajectories or nonlinear PDEs about solitary waves, where V is a model for the tangent space of the phase space. These lectures discuss the theory behind such equations and the implementation of numerical algorithms for their integration. Applications of the theory to the stability of solitary waves, solution of boundary value problems, and hydrodynamic stability are presented.
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Geometric numerical integration for breaking water waves
The widely used governing equations for modelling water waves are Hamiltonian, and therefore one would expect that symplectic integrators would be appropriate for time integration. In this lecture the use of symplectic or other geometric integrators for water waves is discussed. For a simple free surface (a graph) the Hamiltonian formulation is canonical, but still the use of symplectic integrators is not straightforward. For general surfaces, for example breaking waves, one needs a coordinate-free Hamiltonian formulation, and this was first proposed by Benjamin & Olver (1982). For breaking waves the Hamiltonian structure is no longer canonical, and new ideas from geometric integration are needed.
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Multisymplectic structures and geometric integration
In these lectures, the concept of multisymplecticity and its role in the discretization of partial differential equations is discussed. The topics to be discussed include: overview of multisymplecticity, variational integrators and the Cartan form, continuous and discrete conservation of symplecticity, discrete multisymplectic structures, and implications for waves. A new approach to multisymplectic structures will be also introduced and its implications for numerics discussed. The latter idea is based on the observation that any Riemannian manifold has a natural coordinate-free multisymplectic structure on the total exterior algebra bundle, and this ``canonical multisymplectic structure'' turns out to be useful for analysis and numerics of Hamiltonian PDEs.
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