Brynjulf Owren

Computations in a free Lie algebra

Many numerical algorithms involve computations in Lie algebras, like composition and splitting methods, methods involving the Baker–Campbell–Hausdorff formula and the recently developed Lie group methods for integration of differential equations on manifolds. This paper is concerned with complexity and optimization of such computations in the general case where the Lie algebra is free, i.e. no specific assumptions are made about its structure. It is shown how transformations applied to the original variables of a problem yield elements of a graded free Lie algebra whose homogeneous subspaces are of much smaller dimension than the original ungraded one. This can lead to substantial reduction of the number of commutator computations. Witts formula for counting commutators in a free Lie algebra is generalized to the case of a general grading. This provides good bounds on the complexity. The interplay between symbolic and numerical computations is also discussed, exemplified by the new MATLAB toolbox ‘DIFFMAN’

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.

Derivation of efficient, continuous, explicit Runge-Kutta methods

Continuous, explicit Runge–Kutta methods with the minimal number of stages are considered. These methods are continuously differentiable if and only if one of the stages is the FSAL evaluation. A characterization of a subclass of these methods is developed for orders 3, 4, and 5. It is shown how the free parameters of these methods can be used either to minimize the continuous truncation error coefficients or to maximize the stability region. As a representative for these methods the fifth-order method with minimized error coefficients is chosen, supplied with an error estimation method, and analysed by using the DETEST software. The results are compared with a similar implementation of the Dormand–Prince 5(4) pair with interpolant, showing a significant advantage in the new method for the chosen problems.

Topics in structure-preserving discretization

We develop the theory of mixed finite elements in terms of special inverse systems of complexes of differential forms, defined over cellular complexes. Inclusion of cells corresponds to pullback of forms. The theory covers for instance composite piecewise polynomial finite elements of variable order over polyhedral grids. Under natural algebraic and metric conditions, interpolators and smoothers are constructed, which commute with the exterior derivative and whose product is uniformly stable in Lebesgue spaces. As a consequence we obtain not only eigenpair approximation for the Hodge-Laplacian in mixed form, but also variants of Sobolev injections and translation estimates adapted to variational discretizations.

Multi-symplectic integration of the Camassa-Holm equation

The Camassa-Holm equation is rich in geometric structures, it is completely integrable, bi-Hamiltonian, and it represents geodesics for a certain metric in the group of diffeomorphism. Here two new multi-symplectic formulations for the Camassa-Holm equation are presented, and the associated local conservation laws are shown to correspond to certain well-known Hamiltonian functionals. The multi-symplectic discretisation of each formulation is exemplified by means of the Euler box scheme. Numerical experiments show that the schemes have good conservative properties, and one of them is designed to handle the conservative continuation of peakon-antipeakon collisions.

Order barriers for continuous explicit Runge-Kutta methods

In this paper we deal with continuous numerical methods for solving initial value problems for ordinary differential equations, the need for which occurs frequently in applications. Whereas most of the commonly used multistep methods provide continuous extensions by means of an interpolant which is available without making extra function evaluations, this is not always the case for one-step methods. We consider the class of explicit Runge-Kutta methods and provide theorems used to obtain lower bounds for the number of stages required to construct methods of a given unifonn order p. These bounds are similar to the Butcher barriers known for the discrete case, and are derived up to order p = 5. As far as we know, the examples we present of 8-stage continuous Runge-Kutta methods of uniform order 5 are the first of their kind.

Integration methods based on canonical coordinates of the second kind

Summary. We present a new class of integration methods for differential equations on manifolds, in the framework of Lie group actions. Canonical coordinates of the second kind is used for representing the Lie group locally by means of its corresponding Lie algebra. The coordinate map itself can, in many cases, be computed inexpensively, but the approach also involves the inversion of its differential, a task that can be challenging. To succeed, it is necessary to consider carefully how to choose a basis for the Lie algebra, and the ordering of the basis is important as well. For semisimple Lie algebras, one may take advantage of the root space decomposition to provide a basis with desirable properties. The problem of ordering leads us to introduce the concept of an admissible ordered basis (AOB). The existence of an AOB is established for some of the most important Lie algebras. The computational cost analysis shows that the approach may lead to more efficient solvers for ODEs on manifolds than those based on canonical coordinates of the first kind presented by Munthe-Kaas. Numerical experiments verify the derived properties of the new methods.

B-series and Order Conditions for Exponential Integrators

We introduce a general format of numerical ODE-solvers which include many of the recently proposed exponential integrators. We derive a general order theory for these schemes in terms of