Third-order sparse grid generalized spectral elements on hexagonal cells for uniform-speed advection in a plane

Abstract

This paper investigates sparse grids on a hexagonal cell structure using a Local-Galerkin method (LGM) or generalized spectral element method (SEM). Such methods allow sparse grids to be used, known as serendipity grids in square cells. This means that not all points of the full grid are used. Using a high-order polynomial, some points of each cell are eliminated in the discretization, and thus saving Central Processing Unit (CPU) time. Here a sparse SEM scheme is proposed for hexagonal cells. It uses a representation of fields by second-order polynomials and achieves third-order accuracy. As SEM, LGM is strictly local for explicit time integration. This makes LGM more suitable for multiprocessing computers compared with classical Galerkin methods. The computer time depends on the possible timestep and program implementation. Assuming that these do not change when going to a sparse grid, the potential saving of computer time due to sparseness is 1:2. The projected CPU saving in 3-D from sparseness is by a factor of 3:8. A new spectral procedure is used in this paper, called the implied spectral equation (ISE). This procedure allows for some collocation points to use any finite difference scheme of high order and the time derivatives of other spectral coefficients are implied.