Hamiltonian Particle-mesh Method Research Articles

Convergence of the Hamiltonian particle-mesh method for barotropic fluid flow

We prove convergence of the Hamiltonian Particle-Mesh (HPM) method, initially proposed by J. Frank, G. Gottwald, and S. Reich, on a pe- riodic domain when applied to the irrotational shallow water equations as a prototypical model for barotropic compressible uid ow. Under appropriate assumptions, most notably suciently fast decay in Fourier space of the global smoothing operator, and a Strang{Fix condition of order 3 for the local par- tition of unity kernel, the HPM method converges as the number of particles tends to innity and the global interaction scale tends to zero in such a way that the average number of particles per computational mesh cell remains con- stant and the number of particles within the global interaction scale tends to innity. The classical SPH method emerges as a particular limiting case of the HPM algorithm and we nd that the respective rates of convergence are comparable under suitable assumptions. Since the computational complexity of bare SPH is algebraically superlinear and the complexity of HPM is logarithmically su- perlinear in the number of particles, we can interpret the HPM method as a fast SPH algorithm.

Statistical relevance of vorticity conservation in the Hamiltonian particle-mesh method

We conduct long-time simulations with a Hamiltonian particle-mesh method for ideal fluid flow, to determine the statistical mean vorticity field of the discretization. Lagrangian and Eulerian statistical models are proposed for the discrete dynamics, and these are compared against numerical experiments. The observed results are in excellent agreement with the theoretical models, as well as with the continuum statistical mechanical theory for ideal fluid flow developed by Ellis et al. (2002) [10]. In particular the results verify that the apparently trivial conservation of potential vorticity along particle paths within the HPM method significantly influences the mean state. As a side note, the numerical experiments show that a nonzero fourth moment of potential vorticity can influence the statistical mean.

Analysis of a regularized, time‐staggered discretization method and its link to the semi‐implicit method

AbstractA key aspect of the recently proposed Hamiltonian particle‐mesh (HPM) method is its time‐staggered discretization combined with a regularization of the continuous governing equations. In this article, the time discretization aspect of the HPM method is analysed for the linearized, rotating, shallow‐water equations with orography, and the combined effect of time‐staggering and regularization is compared analytically with the popular two‐time‐level semi‐implicit time discretization of the unregularized equations. It is found that the two approaches are essentially equivalent, provided the regularization parameter is chosen appropriately in terms of the time step Δt. The article treats space as a continuum and, hence, its analysis is not limited to the HPM method. Copyright © 2005 Royal Meteorological Society

Hamiltonian Particle-Mesh Method for Two-Layer Shallow-Water Equations Subject to the Rigid-Lid Approximation

We develop a particle-mesh method for two-layer shallow-water equations subject to the rigid-lid approximation. The method is based on the recently proposed Hamiltonian particle-mesh (HPM) method and the interpretation of the rigid-lid approximation as a set of holonomic constraints. The suggested spatial discretization leads to a constrained Hamiltonian system of ODEs which is integrated in time using a variant of the symplectic SHAKE/RATTLE algorithm. It is demonstrated that the elimination of external gravity waves by the rigid-lid approximation can be achieved in a computationally stable and efficient way.

Geometric integration of a wave-vortex model

We introduce a Hamiltonian particle-mesh method for two-dimensional advection under incompressible flow fields. The method is applied to a simplified shallow-water model, called the weak-wave model, which combines slow nonlinear vortical motion with fast linear wave propagation. The advantages of the conservative particle-mesh method are demonstrated by means of the adiabatic energy exchange between vortical and wave motion. More generally, the proposed method is applicable to stable and efficient long-time simulations of other simplified geophysical fluid models.