<LI><B>Liz Mansfield (Kent):</B> <I><FONT face=Times>Noether's theorem for smooth, finite difference and finite element models.</FONT></I>
<P>A key physical property of a physical model with a Lagrangian, that a geometric integrator might emulate, are the conservation laws that arise from symmetries of the Lagrangian. These include conservation of energy, which arises when the Lagrangian is invariant with respect to translation in time, linear momenta (translation with respect to independent variables), angular momenta (rotations with respect to independent variables), and so on. One problem to solve is how a smooth group action carries over to a discretised space. Another is the actual calculation of the conserved quantities.
<P>I shall talk generally about the issues involved. My immediate conclusion will be that the key to solving the problem is to keep the underlying algebraic constructions for discrete models in strict alignment with those of the smooth. In this way, whether a system is inherently discrete or a discretisation of some kind, variational systems, their symmetries and their conservation laws can be studied in a clear, coherent and rigorous way.
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<LI><B>Clyde Martin (Texas Tech.):</B>
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<LI><B>Hans Munthe-Kaas (Bergen):</B> <I><FONT face=Times>Symmetries and the efficiency of Lie group integrators.</FONT></I>
<P>The first half of this talk is a general introduction to work over the recent years on optimizing the performance of Lie group integrators; by minimizing the number of commutators, efficient computation of matrix exponentials and employing efficiently computable local coordinates on Lie groups.
<P>In the second half of the talk we will present recent work on the applications of non-commutative Fourier analysis in the computation of matrix exponentials. This has applications to Lie group integrators for physical problems with domain symmetries, or approximate domain symmetries.
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<LI><B>Marcel Oliver (IUB):</B> <I><FONT face=Times>Numerical evaluation of the Evans function by Magnus integration.</FONT></I>
<P>We use Magnus methods to compute the Evans function for spectral problems as arise when determining the linear stability of travelling wave solutions to reaction-diffusion and related partial differential equations. In a typical application scenario, we need to repeatedly sample the solution to a system of linear non-autonomous ordinary differential equations for different values of one or more parameters as we detect and locate the zeros of the Evans function in the right half of the complex plane.
<P>In this situation, a substantial portion of the computational effort---the numerical evaluation of the iterated integrals which appear in the Magnus series---can be performed independent of the parameters and hence needs to be done only once. More importantly, for any given tolerance Magnus integrators possess lower bounds on the step size which are uniform across large regions of parameter space and which can be estimated \emph{a priori}. We demonstrate, analytically as well as through numerical experiment, that these features render Magnus integrators extremely robust and, depending on the regime of interest, efficient in comparison with standard ODE solvers.
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<LI><B>Brynjulf Owren (NTNU):</B> <I><FONT face=Times>Applications of Lie group integrators and exponential schemes. </FONT></I>
<P>In the first part of this talk, we address the most important applications of Lie group integrators up to date, which we choose to divide into three main cases.
<P>1. Linear problems. Schemes based on Magnus series and variants based on the same principle. These schemes are important in solving and understanding highly oscillatory problems. In PDEs the schemes have been applied with success to the linear Schrodinger equation.
<P>2. Nonlinear problems on manifolds. In principle, all sorts of nonlinear Lie group integrators are applicable here, as for instance the Runge-Kutta-Munthe-Kaas schemes. In some applications there are manifolds with a <I><FONT face=Times>natural</FONT></I> group action which ensures that the numerical approximation remains on some submanifold of Rn. The predominant cases in applications are when the acting group is the orthogonal group, and the manifold is either the orthogonal group, a Stiefel manifold or a sphere.
<P>3. Problems which benfit from the principle of <I><FONT face=Times>curved path building blocks</FONT></I>. These problems are in many cases formulated on Euclidean space, there is no underlying nonlinear manifold. But the use of Lie group integrators allow basic movements that are better suited to the problem at hand. One may use this to deal with stiffness or high oscillations, or simply to obtain high accuracy with a large stepsize. This is similar to the effects seen in highly oscillatory linear problems solved with Magnus series schemes. In the PDE setting, it is popular to use Lie group integrators where the action is by the affine group, and this leads to schemes which are very similar to the exponential integrators first developed by Certaine, Lawson and Norsett.
<P>In the second part of the talk I will focus on some recent developments in exponential integrators. The classical order theory seems straightforward on the outset, but there are particular issues that need to be addressed for exponential integrators. We introduce a general format of schemes which seem to include the majority of the known exponential integrators, as those derived from the Lie group scheme formalism, and the ones recently proposed by Cox and Matthews, and Krogstad. One of the hidden features of exponential integrators is the underlying function spaces from where to choose the coefficient functions of the schemes. Some theorems will be presented which give sharp lower bounds for the dimensions of these spaces. The results will significantly simplify the construction of exponential integrators, to illustrate this we present a few new schemes of high order.
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<LI><B>Claudia Wulff (Surrey):</B> <I><FONT face=Times>Numerical Continuation of Symmetric Periodic Orbits.</FONT></I>
<P>The bifurcation theory and numerics of periodic orbits of general dynamical systems is well-developed, and in recent years there has been a rapid progress in the development of a bifurcation theory for dynamical systems with structure, like symmetry or symplecticity. But there are hardly any results on the numerical computation of those bifurcations yet. In this talk we will show spatio-temporal symmetries of periodic orbits can be exploited numerically, we will describe methods for the computation of symmetry-breaking bifurcations for free group actions and will show how bifurcations increasing the spatiotemporal symmetry (period-halving bifurcations and Hopf bifurcations) can be detected numerically. We will moreover present a method for the numerical continuation of non-degenerate Hamiltonian relative periodic orbits with regular drift-momentum pair. We will apply our methods to coupled cells and present a new family of rotating choreographies of the 3-body system. Our numerical algorithm is based on a multiple shooting algorithm for the numerical computation of periodic orbits via an adaptive Poincare-section and a pathfollowing algorithm using a tangential continuation method with implicit reparametrization. (Joint work with A. Schebesch (Freie Universitat Berlin))
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<LI><B>Antonella Zanna (Bergen):</B> <I><FONT face=Times>The discrete Moser--Veselov algorithm for the rigid body.</FONT></I>
<P>In this talk we consider the discrete Moser--Veselov algorithm for the reduced equations of the rigid body. Moser and Veselov derived this second order algorithm and proved its integrability by using certain matrix polynomial factorizations first introduced by Peter Lax for the KdV equations and other isospectral flows. By backward error analysis, we show that, in the 3x3 case, the discrete Moser--Veselov algorithm is a time-reparametrization of the continuous rigid body equations. By determining some parameters, depending on the Hamiltonian and the other constants of the problem, we construct new higher order, explicit, symplectic and integrable algoritithms for the 3x3 rigid body. </P></LI> |
发表于 2005-7-20 08:13
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