SLIC: A Semi-Lagrangian Implicitly Corrected method for solving the compressible Euler equations

The Semi-Lagrangian (SL) scheme is a popular method used in weather and climate models due to its ability to handle the large Courant numbers encountered in polar regions and jet-streams. This makes SL-based models highly efficient since relatively large time steps can be employed especially when coupled with a semi-implicit scheme for treating the fast acoustic and gravity waves. For problems where the velocity field is unknown in advance and is part of the solution either an extrapolation of the wind or an iterative method is needed to obtain an accurate estimate of the trajectory. This paper proposes an alternative hybrid Semi-Lagrangian-Eulerian scheme whereby the semi-Lagrangian part of the trajectory and interpolation to departure points are performed explicitly and only once per time step with a known velocity field together with a relatively small implicit Eulerian correction. Unlike classical iterative SL schemes, the present method removes (SL) advection completely from the iterative process. We also show that the proposed scheme can also be used in several novel ways including: solving some singular problems associated with classical SL and accuracy issues near solid boundaries. The present approach can also be used as an efficient and straightforward splitting strategy. Crown Copyright (C) 2020 Published by Elsevier Inc. All rights reserved.