A Class of Single-Cell High-Order Semi-Lagrangian Advection Schemes

Abstract

A class of semi-Lagrangian schemes has been derived for solving the advection equation. Compared with other semi-Lagrangian-type schemes, the presented schemes require less computational stencils for interpolation construction. Besides the dependent variable itself, its spatial derivatives are also evaluated based on a Lagrangian invariant solution. This makes estimating the first-order derivatives from the values of the dependent variable at neighboring grid points unnecessary. The resulting numerical formula appears spatially compact and only one mesh cell is needed for constructing the interpolation profile. The 2D basic formulation is based on a rational interpolation function. It shows an ability to prevent numerical oscillation. Some variants of the scheme can be easily obtained by minor modifications. It is easy to get the desired numerical properties such as diffusion reduction, oscillation suppression, or (more strongly) monotonicity with the presented schemes. Grid refinement analysis shows that all the schemes presented in this paper have convergence factors larger than 2 based on an l2 norm. The presented schemes need some extra memory space to store the derivatives of the interpolation function, but do appear competitive with other conventional semi-Lagrangian methods based on Hermite interpolants, in terms of arithmetic operation counts. Parallel implementation shows that the presented schemes are easily portable to a parallel environment with distributed memory architecture and data communications take place only on the cells on the boundaries of the parallel partition.