@article{ShinReichFrank2012,
  author    = {Shin, Seoleun and Reich, Sebastian and Frank, Jason},
  title     = {Hydrostatic Hamiltonian particle-mesh (HPM) methods for atmospheric modelling},
  series = {Quarterly journal of the Royal Meteorological Society},
  volume    = {138},
  journal   = {Quarterly journal of the Royal Meteorological Society},
  number    = {666},
  publisher = {Wiley-Blackwell},
  address   = {Hoboken},
  issn      = {0035-9009},
  doi       = {10.1002/qj.982},
  pages     = {1388 -- 1399},
  year      = {2012},
  abstract  = {We develop a hydrostatic Hamiltonian particle-mesh (HPM) method for efficient long-term numerical integration of the atmosphere. In the HPM method, the hydrostatic approximation is interpreted as a holonomic constraint for the vertical position of particles. This can be viewed as defining a set of vertically buoyant horizontal meshes, with the altitude of each mesh point determined so as to satisfy the hydrostatic balance condition and with particles modelling horizontal advection between the moving meshes. We implement the method in a vertical-slice model and evaluate its performance for the simulation of idealized linear and nonlinear orographic flow in both dry and moist environments. The HPM method is able to capture the basic features of the gravity wave to a degree of accuracy comparable with that reported in the literature. The numerical solution in the moist experiment indicates that the influence of moisture on wave characteristics is represented reasonably well and the reduction of momentum flux is in good agreement with theoretical analysis.},
  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}
}