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Organizers |
G-symplectic general linear methods
by
John Butcher
University of Auckland
A Runge-Kutta method (A, b, c) with the property that diag(b)A+AT diag(b)=bbT is said to be canonical or symplectic. Such methods have an important role in the solution of Hamiltonian problems and for problems possessing a quadratic invariant. Although it is believed that genuine multivalue methods cannot possess an equivalent property, it will be shown that G-symplectic general linear methods can give excellent results.
Date received: October 31, 2007
Symplectic Methods with Transformations
by
Yousaf Habib
Department of Mathematics, University of Auckland
Hamiltonian mechanics is a reformulation of classical mechanics invented by Hamilton (1833). In Hamiltonian mechanics, the equations of motion are based on generalised co-ordinates qi and generalised momenta pi. The Hamiltonian H is a function of p=(p1, p2, , , , , pn) and q=(q1, q2, , , , , qn) and defines the differential equation system,
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Let fH(t, t0) denote the solution operator of the Hamiltonian system.
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Pioneering work in this regard is due to Ruth (1983) and Feng (1985). Later, Sanz-Serna (1988) and Suris (1988) systematically developed symplectic Runge-Kutta methods. Their idea is based on features of algebraic stability introduced, in connection with stiff systems, by Burrage and Butcher (1979) and Crouzeix (1979).
A Runge-Kutta method of order s is symplectic if the coefficients [a, b, c] of the Runge-Kutta method satisfy
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Thus symplectic methods can be found by imposing the condition stated above in addition to other requirements such as order and stability. The left hand side of this euqtion represents a matrix involving the co-efficients of Runge-Kutta method which we call M.
We can use the matrix M to construct a class of Runge Kutta methods which are symplectic by construction. The idea is to pre and post multiply the matrix M of symplectic condition by a Vandermonde matrix. This will give us a system of equations involving the coefficients of Runge-Kutta method. These equation are the order conditions for a class of Runge-Kutta methods. We employ interpolation to evaluate the coefficients of a symplectic Runge-Kutta method.
Date received: November 11, 2007
Stability of Numerical Solvers for Ordinary Differential Equations
by
Allison Heard
University of Auckland
Coauthors: John Butcher
Using the example of the second order BDF method, I will consider the stablity of numerical methods used with variable stepsize, and how this depends not only on the method used but also its formualation. The `scale and modify' approach, introduced by J Butcher and Z Jackiewicz, can be used to extend the stability region. This technique will be described with reference to the underlying one-step method.
Date received: October 31, 2007
Butcher trees and curve search in nonlinear optimization
by
Laurent O. Jay
Department of Mathematics, University of Iowa, USA
Coauthors: Darin G. Mohr (Department of Mathematics, University of Iowa, USA)
In this talk we show that the field of nonlinear optimization may benefit from techniques developed primarily for the numerical integration of ordinary differential equations (ODEs). Here we are more specifically concerned with improving line search methods to new curve search methods for problems in unconstrained nonlinear optimization. For line search methods a search direction is computed and then a line search is done on the corresponding half-line. The main new idea is to obtain at each step the parametrization of a desired nonlinear geometric curve with better minimization properties for small values of the steplength and then to apply a curve search. Desired geometric curves can be determined thanks to a careful analysis based on Butcher trees, this is our first connection to the numerical integration of ODEs. We approximate a desired geometric curve using methods analogous to Runge-Kutta methods, this is our second connection to the numerical integration of ODEs. Numerical methods for ODEs applied to the gradient flow of an objective function have been considered in the past by several authors in nonlinear optimization. We show in particular that the gradient flow from a point corresponds to a geometric curve which is generally not the most desirable geometric curve for small values of the steplength.
Date received: October 31, 2007
Homogeneous Variational Integrators for Lagrangian Dynamics on Two-Spheres
by
Melvin Leok
Purdue University
Coauthors: Taeyoung Lee, and N. Harris McClamroch
Homogeneous variational integrators for Lagrangian flows on two-spheres are constructed by lifting the variational principle on S2 to a constrained variational principle on SO(3), through the use of constrained variations which quotient out the local isotropy subgroup of the action of SO(3) on S2. This is analogous to the reconstruction process in reduction theory.
This approach yields compact expressions for the continuous and discrete dynamics of mechanisms consisting of particles with inter-particle length constraints. These provide the basis for constructing geometrically exact numerical schemes for representing flexible structures and surfaces arising in modern engineering applications.
This research is partially supported by NSF grant DMS-0714223 and DMS-0714223.
Date received: October 22, 2007
Achieving Brouwer's law of round-off error
by
Robert McLachlan
Massey University
Coauthors: Ernst Hairer
In 1937 the astronomer Dirk Brouwer suggested that round-off errors in the numerical solution of differential equations should be independent random variables with mean zero, so that their cumulative effect would be that of a random walk. However, standard implementations of implicit Runge-Kutta methods do not obey this law, instead showing a much more rapid and systematic error growth. I will explain Ernst Hairer's and my attempts to understand and correct this problem, which may have implications for long-term simulations of the solar system.
Date received: October 31, 2007
On explicit adaptive symplectic integration of separable Hamiltonian systems
by
Klas Modin
Numerical Analysis, Lund University
Coauthors: Gustaf Söderlind
It is well known that symplecticity is preserved under Sundman transformations if and only if the time scaling function is a first integral of the flow. This observation, in conjunction with Hamiltonian splitting methods, allows the construction of explicit adaptive symplectic methods for a commonly used class of scaling functions.
Due to symplecticity these adaptive integrators have excellent long time energy behavior, which is theoretically explained using standard results on the existence of a modified Hamiltonian function. Contrary to reversible adaptive integration, the constructed methods have good long time behavior also for non-reversible and/or non-integrable systems.
Comparisons between reversible adaptive methods and symplectic adaptive methods are given by several numerical examples.
Date received: October 31, 2007
Geometric integration, high oscillation and resonance.
by
Dion O'Neale
Massey University, Palmerston North.
Coauthors: Robert McLachlan
We look at the performance of trigonometric integrators applied to highly oscillatory differential equations. It is widely known that some of the trigonometric integrators suffer from low order resonances for particular step sizes.
We show here that, in general, trigonometric integrators also suffer from higher order resonances which can lead to loss of nonlinear stability. We illustrate this with the Fermi-Pasta-Ulam problem, a highly oscillatory Hamiltonian system.
We also show that in some cases trigonometric integrators preserve invariant or adiabatic quantities but at the wrong values. We use statistical properties such as time averages to further evaluate the performance of the trigonometric methods and compare the performance with that of the mid-point rule.
Date received: October 30, 2007
Geometric Integration of Ordinary Differential Equations
by
Reinout Quispel
La Trobe University, Australia
Coauthors: David McLaren
Geometric integration is the numerical integration of a differential equation, while preserving one or more of its geometric/physical properties exactly, i.e. to within round-off error. Many of these geometric properties are of crucial importance in physical applications: preservation of energy, momentum, angular momentum, phase-space volume, symmetries, time-reversal symmetries, symplectic structure and dissipation are examples. In this talk we present a survey of geometric numerical integration methods for ordinary differential equations, focusing on symplectic methods resp. energy-preserving methods. Our aim has been to make the review of use for a general mathematical audience.
Date received: August 21, 2007
Adaptive Geometric Integration: Structural Aspects of Reversible Step Size Control
by
Gustaf Söderlind
Lund University, Sweden
Adaptive techniques are of great importance in the numerical solution of differential equations. A smaller number of grid points will suffice if they are properly located. However, in some applications, e.g. in integrable Hamiltonian systems, it is important to preserve invariants of the analytic solution. This imposes structural constraints on step size control algorithms. These constraints are explored in terms of commutative diagrams, and it is shown that if Y is the step size map, then -Y must be an involution for time reversibility to be preserved in the discrete system. Finally, we will briefly look at Sundman transformations and construct a nonlinear Hamiltonian control system to make the Störmer-Verlet method adaptive, while perserving time symmetry, reversibility and the long-term behaviour normally only associated with constant step sizes.
Date received: December 2, 2007
Evaluating Performance of Exponential Integrators.
by
Mayya Tokman
University of California, Merced
A number of exponential integrators have been proposed in the recent years as an alternative to standard schemes for solving large stiff systems of ODEs. A thorough study of the performance of different exponential integrators as well as computational savings they offer for a variety of applications still remains to be carried out. We discuss a class of exponential propagation iterative methods (EPI) and compare them with other exponential integrators. This presentation focuses on construction of these schemes and discussion of their properties as compared to implicit, explicit methods and other exponential integrators. Several exponential integrators as well as implicit and explicit methods will be compared and their performance evaluated using demonstrative numerical examples.
Date received: November 2, 2007
The efficient evaluation of functions related to the matrix exponential
by
Will Wright
La Trobe University
Recent interest in the class of exponential integrators has led to the need for the efficient evaluation of the matrix exponential and related functions. Exponential integrators are typically employed on discretized PDEs, which often have a very large number of differential equations. Therefore, it is generally unfeasible to compute the matrix exponential or the related functions but only their action on a vector. We will outline our implementation which is based on the Krylov subspace approach.
Date received: November 20, 2007
Dynamics and Numerics of some generalised Euler equations
by
Philip Zhang
Massey University
Coauthors: Robert McLachlan
Since V.I. Arnold proposed a geometrical approach to Euler fluid equations in 1966, much attention has been attracted to the generalised Euler equations (or Euler-Poincaré equations), which stand for the geodesic equations on some Lie groups. Misiolek et al proved recently that the famous KdV and Camassa-Holm equations are the generalised Euler equations on the Bott-Virasoro group with respect to the L2 metric and H0 metric respectively. In this talk, we will investigate the dynamics of the generalised Euler equations on the Bott-Virasoro group with respect to the general Hk metric. Some wellposedness and numeric results will be given.
Date received: November 11, 2007