A Semi-implicit Semi-lagrangian Shallow-water Model for Massively Parallel Processors

The formulation of a grid point shallow-water model proposed by C^ ot e and Stan-iforth is described. The model employs a hybrid nite-volume//nite-element spatial discretisation on a staggered Arakawa B type grid, where the geopoten-tial is computed at grid points and the wind u is found at grid cell centers. Time integration is based upon a two-time-level semi-implicit, semi-Lagrangian scheme. The nonlinear Helmholtz problem obtained by algebraic elimination of the time-discretised equations is solved by xed-point iteration. Direct methods are rst considered for solution of the embedded linear Helmholtz kernel. A fast tensor product solver has been implemented which employs data transposition to perform FFT computations in the x-direction followed by decoupled tridiagonal solves in the y-direction. To achieve scaled speed-ups for reasonable grid sizes on each processor, the communication costs associated with semi-Lagrangian advection and the elliptic solver must be minimized. Algorithms for parallel advection are reviewed and several alternatives for MIMD computation are discussed. Transposition of grid point data during each nonlinear iteration imposes a heavy communications overhead on the direct elliptic solver. Thus, iterative methods are proposed as a viable alternative.