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  • 1. Celledoni, Elena
    et al.
    Kingsley Kometa, Bawfeh
    Verdier, Olivier
    Institutt for matematiske fag, NTNU, Trondheim, Norway.
    High order semi-Lagrangian methods for the incompressible Navier–Stokes equations2016In: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 66, no 1, p. 91-115Article in journal (Refereed)
    Abstract [en]

    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.

  • 2.
    McLachlan, Robert
    et al.
    Institute of Fundamental Sciences, Massey University, Palmerston North, New Zealand.
    Modin, Klas
    Department of Mathematical Sciences, Chalmers University of Technology, Gothenburg, Sweden.
    Verdier, Olivier
    Department of Mathematics, University of Bergen, Bergen, Norway.
    Collective Lie–Poisson integrators on R³2014In: IMA Journal of Numerical Analysis, ISSN 0272-4979, E-ISSN 1464-3642Article in journal (Refereed)
    Abstract [en]

    We develop Lie–Poisson integrators for general Hamiltonian systems on R³ equipped with the rigid body bracket. The method uses symplectic realisation of RR³ on T*R² and application of symplectic Runge–Kutta schemes. As a side product, we obtain simple symplectic integrators for general Hamiltonian systems on the sphere S².

  • 3.
    McLachlan, Robert
    et al.
    Institute of Fundamental Sciences, Massey University, Palmerston North, New Zealand.
    Modin, Klas
    Department of Mathematical Sciences, Chalmers University of Technology, Gothenburg, Sweden.
    Verdier, Olivier
    Department of Mathematics, University of Bergen, Norway.
    Collective symplectic integrators2014In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 27, no 6, p. 1525-1542Article in journal (Refereed)
    Abstract [en]

    We construct symplectic integrators for Lie-Poisson systems. The integrators are standard symplectic (partitioned) Runge-Kutta methods. Their phase space is a symplectic vector space equipped with a Hamiltonian action with momentum map J whose range is the target Lie-Poisson manifold, and their Hamiltonian is collective, that is, it is the target Hamiltonian pulled back by J. The method yields, for example, a symplectic midpoint rule expressed in 4 variables for arbitrary Hamiltonians on so(3)*. The method specializes in the case that a sufficiently large symmetry group acts on the fibres of J, and generalizes to the case that the vector space carries a bifoliation. Examples involving many classical groups are presented.

  • 4. Munthe-Kaas, Hans
    et al.
    Verdier, Olivier
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Aromatic Butcher series2016In: Foundations of Computational Mathematics, ISSN 1615-3375, E-ISSN 1615-3383, Vol. 16, no 1, p. 183-215Article in journal (Refereed)
    Abstract [en]

    We show that without other further assumption than affine equivariance and locality, a numerical integrator has an expansion in a generalized form of Butcher series (B-series), which we call aromatic B-series. We obtain an explicit description of aromatic B-series in terms of elementary differentials associated to aromatic trees, which are directed graphs generalizing trees. We also define a new class of integrators, the class of aromatic Runge-Kutta methods, that extends the class of Runge-Kutta methods and have aromatic B-series expansion but are not B-series methods. Finally, those results are partially extended to the case of more general affine group equivariance.

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