Forward semi-Lagrangian schemes for advection and gravity waves

Several semi-Lagrangian schemes are designed for application to problems of advection and gravity waves in modeling fluid dynamics. A short-wave filter is also designed to eliminate spurious short waves and to achieve positive-definiteness. These numerical schemes employ forward trajectories instead of the traditional backward trajectories to receive advantages in accuracy and efficiency. The technical difficulty of interpolation from the irregularly spaced Lagrangian grid to the regularly spaced Eulerian grid is overcome by the discoveries of a split method and an internet method. The one-dimensional advection scheme and the short-wave filter constitute the foundation of these schemes. Using the split interpolation method, we extend the one-dimensional advection scheme to a two-dimensional one that is simple, efficient and accurate. The great efficiency of the split method is accompanied by a restriction which is slightly more stringent than the standard semi-Lagrangian criterion. Using the internet interpolation method, we obtain a general two-dimensional advection scheme which has virtually no restrictions. The general advection scheme is therefore applicable to problems of any extent of deformation with arbitrary Courant numbers. The extension of these advection schemes to coupled sets of equations is attained by using a tensor transformation to evaluate the prognostic variables on the Lagrangian grid. This idea is demonstrated with a simple forward-backward time-discretization in the context of shallow-water equations. The model is well justified by testing the important process of geostrophic adjustment and the propagation of gravity waves.