Håvard Berland (TBA)
Elena Celledoni "Semi-Lagrangian schemes for convection-dominated flows" (45 min)
We consider semi-Lagrangian methods in the context of spectral-element discretizations of PDEs. A classical approach and a new approach are presented.
The methods are applied to 2D convection problems and compared with Eulerian schemes. The superiority of the Lagrangian schemes over the Eulerian ones in dealing with numerical dispersion is illustrated in the numerical experiments.
Slides of the talk
Bjarte Hægland "Level sets and free surface flows" (45 min)
Stein Krogstad "Accurate 'integrating factor' methods for stiff nonlinear PDEs"(30 min)
We consider integrating factor (IF) related methods for solving stiff nonlinear PDEs where typically the stiffnes lies in the linear part. The standard IF method applied to an system of ODEs arising from a space-discretized PDE, is known to have two serious drawbacks. It has large error coefficients and it does not have the same fixed points as the original system. We propose a simple modification to the IF method with huge improvements in accuracy and where the fixed point property is fulfilled. We comment on stability and present a few numerical experiments.
Christian Lubich, "Variational splitting for multiconfiguration quantum dynamics" (45 min)
A numerical time integration method for the nonlinear partial differential equations of the multiconfiguration time-dependent Hartree (MCTDH) approach to quantum dynamics is proposed and analyzed. The method is based on a splitting of the quantum Hamiltonian, though not on the level of the MCTDH equations but of the underlying variational principle. The integrator suffers no step size restriction caused by the unbounded Laplacian in the kinetic part of the Hamiltonian, in contrast to direct time discretizations of the MCTDH equations.
Link to paper (zipped file)
Robert McLachan "Lie group foliations: Dynamical systems and integrators" (45 min)
Foliate systems are those which preserve some (possibly singular) foliation of phase space, such as systems with integrals, systems with continuous symmetries, and skew product systems. We study numerical integrators which also preserve the foliation. The case in which the foliation is given by the orbits of an action of a Lie group has a particularly nice structure, which we study in detail, giving conditions under which all foliate vector fields can be written as the sum of a vector field tangent to the orbits and a vector field invariant under the group action. This allows the application of many techniques of geometric integration, including splitting methods and Lie group integrators.
Borko Minchev "Spectral Element Method" (30 min)
The spectral element methods (p-version of finite element method) combines the geometric flexibility of finite element methods with the accuracy of conventional spectral methods. The basic idea of this method is to divide the complete domain into several sub domains. I will consider the main framework for spectral element methods in R^d based on tensor-product sum-factorization and discuss some connection with time integration
Per Christian Moan (TBA) 45 min
Hans Munthe-Kaas (TBA) 30 min
Jitse Niesen "Optimizing step size subject to a bound on the global error" (30 min)
The following problem is addressed: suppose we want to solve an ordinary differential equation numerically with a variable step-size method while keeping the global error within a user-specified tolerance, how do we determine the step size (ususally denoted h).
This problem falls within the framework of state-constrained optimal control theory. We discuss the necessary conditions for optimality, which take the form of a boundary value problem with jump conditions, and show that in simple cases, these can be used to derive the optimal step size. However, in general we need to resort to numerics to solve the optimality conditions. We present a simple algorithm, and end with an example.
Slides of the talk (gzipped ps)
Brynjulf Owren "Design and analysis of Lie group integrators -- a modular approach" (45 min)
In this talk we take a new look at Lie group integrators, and consider how such schemes can be broken down into elementary operations acting on various types of objects belonging to: The linear span of frozen vector fields, their Lie algebra closure, and the exponential of these linear spaces. The operations are typically: Freezing of vector fields, forming linear combinations of vector fields, forming commutators between vector fields, exponentiating such vector fields and composing maps on a manifold. Every known Lie group integrator based on exponentials can be built in this way. Next we provide simple recursion formulas which show how B-series of each of the above mentioned objects are transformed by an elementary operation. The formulas are easy to implement with a computer algebra system, and one may design a software tool that can easily provide order conditions for any type of Lie group integrator which is based on these elementary operations.
Slides of the talk
Reinout Quispel "Splitting methods for polynomial vector fields" (45 min)
Bård Skaflestad "Operator Integration Factor Splitting Methods" (30 min)
In 1990 Maday et. al. introduced a framework for generating splitting methods for initial value problems based on integration factors. The idea is that by introducing such factors we can effectively solve the constituent parts of a multicomponent vector field uncoupled, an attractive feature if the total vector field is hard to integrate.The integration factor (a square matrix) is never actually formed, but its action is evaluated by solving associated sub-IVPs. I will present the framework, show some results, discuss what I percive to be a limitation and discribe possible future work.
The talk will be largely based on the 1990 paper
"An Operator-Integration-Factor Splitting Method for Time-Dependent Problems: Application to Incompressible Fluid Flow" by Y. Maday, A.T. Patera and E.M. Rønquist.
Mark Sofroniou, "Hybrid solvers for splitting and composition methods I" (35 min)
Giulia Spaletta, "Hybrid solvers for splitting and composition methods II" (25 min)
Will Wright No talk
Antonella Zanna "On the spectra of certain matrices generated by involutory automorphisms" (45 min)
We investigate the spectral structure of the matrix K = 1/2 (A+HAH), where H is a unitary involution. We characterise the eigenvalues of K as the zeros of a rational function and prove that, for a normal A, the spectrum of K resides in the convex hull of the eigenvalues of A. This need not be true for non-normal matrices.
Slides of the talk