@article{StaniforthWoodReich2006,
  author    = {Staniforth, Andrew and Wood, Nigel and Reich, Sebastian},
  title     = {A time-staggered semi-Lagrangian discretization of the rotating shallow-water equations},
  series = {Quarterly journal of the Royal Meteorological Society},
  volume    = {132},
  journal   = {Quarterly journal of the Royal Meteorological Society},
  number    = {621C},
  publisher = {Wiley},
  address   = {Weinheim},
  issn      = {0035-9009},
  doi       = {10.1256/qj.06.30},
  pages     = {3107 -- 3116},
  year      = {2006},
  abstract  = {A time-staggered semi-Lagrangian discretization of the rotating shallow-water equations is proposed and analysed. Application of regularization to the geopotential field used in the momentum equations leads to an unconditionally stable scheme. The analysis, together with a fully nonlinear example application, suggests that this approach is a promising, efficient, and accurate alternative to traditional schemes.},
  language  = {en}
}
@article{Reich2006,
  author    = {Reich, Sebastian},
  title     = {Linearly implicit time stepping methods for numerical weather prediction},
  series = {BIT : numerical mathematics ; the leading applied mathematics journal for all computational mathematicians},
  volume    = {46},
  journal   = {BIT : numerical mathematics ; the leading applied mathematics journal for all computational mathematicians},
  publisher = {Springer},
  address   = {Dordrecht},
  issn      = {0006-3835},
  doi       = {10.1007/s10543-006-0065-0},
  pages     = {607 -- 616},
  year      = {2006},
  abstract  = {The efficient time integration of the dynamic core equations for numerical weather prediction (NWP) remains a key challenge. One of the most popular methods is currently provided by implementations of the semi-implicit semi-Lagrangian (SISL) method, originally proposed by Robert (J. Meteorol. Soc. Jpn., 1982). Practical implementations of the SISL method are, however, not without certain shortcomings with regard to accuracy, conservation properties and stability. Based on recent work by Gottwald, Frank and Reich (LNCSE, Springer, 2002), Frank, Reich, Staniforth, White and Wood (Atm. Sci. Lett., 2005) and Wood, Staniforth and Reich (Atm. Sci. Lett., 2006) we propose an alternative semi-Lagrangian implementation based on a set of regularized equations and the popular Stormer-Verlet time stepping method in the context of the shallow-water equations (SWEs). Ultimately, the goal is to develop practical implementations for the 3D Euler equations that overcome some or all shortcomings of current SISL implementations.},
  language  = {en}
}
@article{FrankMooreReich2006,
  author    = {Frank, Jason and Moore, Brian E. and Reich, Sebastian},
  title     = {Linear PDEs and numerical methods that preserve a multisymplectic conservation law},
  issn      = {1064-8275},
  doi       = {10.1137/050628271},
  year      = {2006},
  abstract  = {Multisymplectic methods have recently been proposed as a generalization of symplectic ODE methods to the case of Hamiltonian PDEs. Their excellent long time behavior for a variety of Hamiltonian wave equations has been demonstrated in a number of numerical studies. A theoretical investigation and justification of multisymplectic methods is still largely missing. In this paper, we study linear multisymplectic PDEs and their discretization by means of numerical dispersion relations. It is found that multisymplectic methods in the sense of Bridges and Reich [Phys. Lett. A, 284 ( 2001), pp. 184-193] and Reich [J. Comput. Phys., 157 (2000), pp. 473-499], such as Gauss-Legendre Runge-Kutta methods, possess a number of desirable properties such as nonexistence of spurious roots and conservation of the sign of the group velocity. A certain CFL-type restriction on Delta t/Delta x might be required for methods higher than second order in time. It is also demonstrated by means of the explicit midpoint method that multistep methods may exhibit spurious roots in the numerical dispersion relation for any value of Delta t/Delta x despite being multisymplectic in the sense of discrete variational mechanics [J. E. Marsden, G. P. Patrick, and S. Shkoller, Commun. Math. Phys., 199 (1999), pp. 351-395]},
  language  = {en}
}