An Efficient Interpolation Procedure for High-Order Three-Dimensional Semi-Lagrangian Models

Abstract

An efficient method is proposed for performing the grid interpolations required at each advective time step of a multilevel, limited-area semi-Lagrangian model. The distinctive feature of the method is that it is composed of a cascade of one-dimensional interpolations of entire fields of data through a sequence of intermediate grids. These intermediate grids are formed as hybrid combinations of the standard model grid coordinates, which, together with the Lagrangian coordinates, are delineated by the origins of the trajectories that characterize the semi-Lagrangian method. When applied at Nth-order accuracy, the cascade method requires only O(N) operations compared with O(N3) for the conventional three-dimensional interpolation methods, making the adoption of high-order schemes attractive. The technique was tested on a large number (100) of 48-h forecasts and was found to be as accurate as the conventional interpolation procedures based on point-by-point Cartesian products of one-dimensional inter-polators. However, the cascade interpolation technique was 2.9, 6.1, and 10.2 times as fast as the conventional interpolation scheme for fourth-, sixth-, and eighth-order, respectively. We observe that the cascade method is equally applicable to the problem of interpolation from the grid of termini of forward trajectories to the standard model grid, for which there is no obvious counterpart in the Cartesian product method. Our technique therefore opens the way to a whole new class of high-order accurate semi-Lagrangian methods that incorporate the use of forward trajectories as part of the time-stepping process.