A Characteristic-Based Semi-Lagrangian Method for Hyperbolic Systems of Conservation Laws

A characteristic-based semi-Lagrangian (CSL) method is developed for hyperbolic systems of conservation laws. In the CSL method, the governing equations are first transformed into a system of equations in characteristic form and then the Lagrangian form of the transformed equations is solved along the characteristic directions. By definition of hyperbolicity, such a transformation always exists. The CSL method is first illustrated by applying it to the one-dimensional (1-D) shallow water equations to solve the Rossby geostrophic adjustment problem. The authors then apply the CSL method to the 1-D fully-compressible, non-hydrostatic atmospheric equations to solve the hydrostatic adjustment problem. Transient solutions for both the linear and nonlinear problems are obtained and analyzed. It is shown that the CSL method produces more accurate solutions than the conventional semi-implicit semi-Lagrangian (SISL) method. It is also shown that the open boundary conditions can be easily implemented using the CSL method, which provides another advantage of the CSL method for regional atmospheric modeling. The extension to multidimensional hyperbolic systems is discussed and a simple demonstration is presented. The present study indicates that, although the SISL method is commonly used in the atmospheric models, the CSL method is potentially a better choice for fully-compressible, non-hydrostatic atmospheric modeling.