Elena Celledoni

Preserving energy resp. dissipation in numerical PDEs using the Average Vector Field method

We give a systematic method for discretizing Hamiltonian partial differential equations (PDEs) with constant symplectic structure, while preserving their energy exactly. The same method, applied to PDEs with constant dissipative structure, also preserves the correct monotonic decrease of energy. The method is illustrated by many examples. In the Hamiltonian case these include: the sine-Gordon, Korteweg-de Vries, nonlinear Schrodinger, (linear) time-dependent Schrodinger, and Maxwell equations. In the dissipative case the examples are: the Allen-Cahn, Cahn-Hilliard, Ginzburg-Landau, and heat equations.

Lie group methods for rigid body dynamics and time integration on manifolds

Recently there has been an increasing interest in time integrators for ordinary differential equations which use Lie group actions as a primitive in the design of the methods. These methods are usually phrased in an abstract sense for arbitrary Lie groups and actions. We show here how the methods look when applied to the rigid body equations in particular and indicate how the methods work in general. An important part of the Lie group methods involves the computation of a coordinate map and its derivative. Various options are available, and they vary in cost, accuracy and ability to approximately conserve invariants. We discuss how the computation of these maps can be optimized for the rigid body case, and we provide numerical experiments which give an idea of the performance of Lie group methods compared to other known integration schemes.

Commutator-free Lie group methods

We propose a new format of Lie group methods which does not involve commutators and which uses a much lower number of exponentials than those proposed by Crouch and Grossman. By reusing flow calculations in different stages, the complexity is even further reduced. We argue that the new methods may be particularly useful when applied to problems on homogeneous manifolds with large isotropy groups, or when used for stiff problems. Numerical experiments verify these claims when applied to a problem on the orthogonal Stiefel manifold, and to an example arising from the semidiscretization of a linear inhomogeneous heat conduction problem.

Symmetric Exponential Integrators with an Application to the Cubic Schrödinger Equation

Abstract In this article, we derive and study symmetric exponential integrators. Numerical experiments are performed for the cubic Schrödinger equation and comparisons with classical exponential integrators and other geometric methods are also given. Some of the proposed methods preserve the L2-norm and/or the energy of the system.

A Krylov projection method for systems of ODEs

Abstract In this paper we consider approximations of solutions of IVPs obtained through projections into Krylov subspaces. Numerical experiments on parabolic equations illustrate the performance of the method.

A Class of Intrinsic Schemes for Orthogonal Integration

Numerical integration of ODEs on the orthogonal Stiefel manifold is considered. Points on this manifold are represented as n × k matrices with orthonormal columns, of particular interest is the case when

Efficient time-symmetric simulation of torqued rigid bodies using Jacobi elliptic functions

n\gg k

An introduction to Lie group integrators - basics, new developments and applications

. Mainly two requirements are imposed on the integration schemes. First, they should have arithmetic complexity of order nk2. Second, they should be intrinsic in the sense that they require only the ODE vector field to be defined on the Stiefel manifold, as opposed to, for instance, projection methods. The design of the methods makes use of retractions maps. Two algorithms are proposed, one where the retraction map is based on the QR decomposition of a matrix, and one where it is based on the polar decomposition. Numerical experiments show that the new methods are superior to standard Lie group methods with respect to arithmetic complexity, and may be more reliable than projection methods, owing to their intrinsic nature.

The Exact Computation of the Free Rigid Body Motion and Its Use in Splitting Methods

If the three moments of inertia are distinct, the solution to the Euler equations for the free rigid body is given in terms of Jacobi elliptic functions. Using the arithmetic–geometric mean algorithm (Abramowitz and Stegun 1992 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (New York: Dover)), these functions can be calculated efficiently and accurately. Compared to standard numerical ODE and Lie–Poisson solvers, the overall approach yields a faster and more accurate numerical solution to the Euler equations. This approach is designed for mass asymmetric rigid bodies. In the case of symmetric bodies, the exact solution is available in terms of trigonometric functions, see Dullweber et al (1997 J. Chem. Phys. 107 5840–51), Reich (1996 Fields Inst. Commun. 10 181–91) and Benettin et al (2001 SIAM J. Sci. Comp. 23 1189–203) for details. In this paper, we consider the case of asymmetric rigid bodies subject to external forces. We consider a strategy similar to the symplectic splitting method proposed in Reich (1996 Fields Inst. Commun. 10 181–91) and Dullweber et al (1997 J. Chem. Phys. 107 5840–51). The method proposed here is time-symmetric. We decompose the vector field of our problem into a free rigid body (FRB) problem and another completely integrable vector field. The FRB problem consists of the Euler equations and a differential equation for the 3 × 3 orientation matrix. The Euler equations are integrated exactly while the matrix equation is approximated using a truncated Magnus series. In our experiments, we observe that the overall numerical solution benefits greatly from the very accurate solution of the Euler equations. We apply the method to the heavy top and the simulation of artificial satellite attitude dynamics.

On the implementation of Lie group methods on the Stiefel manifold

We give a short and elementary introduction to Lie group methods. A selection of applications of Lie group integrators are discussed. Finally, a family of symplectic integrators on cotangent bundles of Lie groups is presented and the notion of discrete gradient methods is generalised to Lie groups.