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We study geometric properties of Krylov projection methods for large and sparse linear Hamiltonian systems. We consider in particular energy-preservation. We discuss the connection to structure preserving model reduction. We illustrate the performance of the methods by applying them to Hamiltonian PDEs.

No Runge–Kutta method can be energy preserving for all Hamiltonian systems. But for problems in which the Hamiltonian is a polynomial, the averaged vector field (AVF) method can be interpreted as a Runge–Kutta method whose weights bib_i and abscissae cic_i represent a quadrature rule of degree at least that of the Hamiltonian. We prove that when the number of stages is minimal, the Runge–Kutta scheme must in fact be identical to the AVF scheme.

We propose a class of semi-Lagrangian methods of high approximation order in space and time, based on spectral element space discretizations and exponential integrators of Runge–Kutta type. The methods were presented in Celledoni and Kometa (J Sci Comput 41(1):139–164, 2009 ) for simpler convection–diffusion equations. We discuss the extension of these methods to the Navier–Stokes equations, and their implementation using projections. Semi-Lagrangian methods up to order three are implemented and tested on various examples. The good performance of the methods for convection-dominated problems is demonstrated with numerical experiments.

A novel integration method for quadratic vector fields was introduced by Kahan in 1993. Subsequently, it was shown that Kahan's method preserves a (modified) measure and energy when applied to quadratic Hamiltonian vector fields. Here we generalize Kahan's method to cubic resp. higher degree polynomial vector fields and show that the resulting discretization also preserves modified versions of the measure and energy when applied to cubic resp. higher degree polynomial Hamiltonian vector fields.

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 $n\\gg k$. 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.

Calculating cost-effective solutions to particle dynamics in viscous flows is an important problem in many areas of industry and nature. We implement a second-order symmetric splitting method on the governing equations for a rigid spheroidal particle model with torques, drag and gravity. The method splits the operators into a vector field that is conservative and one that takes into account the forces of the fluid. Error analysis and numerical tests are performed on perturbed and stiff particle-fluid systems. For the perturbed case, the splitting method greatly improves the solution accuracy, when compared to a conventional multistep method, and the global error behaves as O ( ε h 2 ) for roughly equal computational cost. For stiff systems, we show that the splitting method retains stability in regimes where conventional methods blow up. In addition, we show through numerical experiments that the global order is reduced from O ( h 2 / ε ) in the perturbed regime to O ( h ) in the stiff regime.

We introduce energy-preserving integrators for nonholonomic mechanical systems. We will see that the nonholonomic dynamics is completely determined by a triple ( D ∗ , Π , H ) , where D ∗ is the dual of the vector bundle determined by the nonholonomic constraints, Π is an almost-Poisson bracket (the nonholonomic bracket) and H : D ∗ → R is a Hamiltonian function. For this triple, we can apply energy-preserving integrators, in particular, we show that discrete gradients can be used in the numerical integration of nonholonomic dynamics. By construction, we achieve preservation of the constraints and of the energy of the nonholonomic system. Moreover, to facilitate their applicability to complex systems which cannot be easily transformed into the aforementioned almost-Poisson form, we rewrite our integrators using just the initial information of the nonholonomic system. The derived procedures are tested on several examples: a chaotic quartic nonholonomic mechanical system, the Chaplygin sleigh system, the Suslov problem and a continuous gearbox driven by an asymmetric pendulum. Their performance is compared with other standard methods in nonholonomic dynamics, and their merits verified in practice.

There are several applications in which one needs to integrate a system of ODEs whose solution is an n×p matrix with orthonormal columns. In recent papers algorithms of arithmetic complexity order np2 have been proposed. The class of Lie group integrators may seem like a worth while alternative for this class of problems, but it has not been clear how to implement such methods with O(np2) complexity. In this paper we show how Lie group methods can be implemented in a computationally competitive way, by exploiting that analytic functions of n×n matrices of rank 2p can be computed with O(np2) complexity.

In this paper we consider the case of nonlinear convection-diffusion problems with a dominating convection term and we propose exponential integrators based on the composition of exact pure convection flows. These methods can be applied to the numerical integration of the considered PDEs in a semi-Lagrangian fashion. Semi-Lagrangian methods perform well on convection dominated problems (Pironneau in Numer. Math. 38:309–332, 1982 ; Hockney and Eastwood in Computer simulations using particles. McGraw-Hill, New York, 1981 ; Rees and Morton in SIAM J. Sci. Stat. Comput. 12(3):547–572, 1991 ; Baines in Moving finite elements. Monographs on numerical analysis. Clarendon Press, Oxford, 1994 ). In these methods linear convective terms can be integrated exactly by first computing the characteristics corresponding to the gridpoints of the adopted discretization, and then producing the numerical approximation via an interpolation procedure.

A novel integration method for quadratic vector fields was introduced by Kahan in 1993. Subsequently, it was shown that Kahan's method preserves a (modified) measure and energy when applied to quadratic Hamiltonian vector fields. Here we generalize Kahan's method to cubic resp. higher degree polynomial vector fields and show that the resulting discretization also preserves modified versions of the measure and energy when applied to cubic resp. higher degree polynomial Hamiltonian vector fields.