Non-iteration dimensional-split semi-Lagrangian advection with Riemann solver in non-hydrostatic system

A non-hydrostatic equation system is used to test non-iteration dimentional-split semi-Lagrangian (NDSL) advection with Riemann invariant characteristic equation (RICE) to resolve acoustic wave explicitly. Since NDSL is unconditional stable for advection, theoretically we can use as large time step as possible, however, RICE includes acoustic speed in the advection, the time step may not be too large as compared to semi-Lagrangian and semi-implicit time integration scheme. Several experiments for this NDSL with RICE will be presented, including one dimension and two dimension experiments and warm bubble test as shown in literature. The preliminary results show that no limitation of time step should be used in one-dimensional test but some limitations in two dimensional tests due to the nature of Riemann solver. Due to the long distance advection by acoustic speed, the forcing works on the advection may have to consider entire trajectory as compared to central mean value with some tests. These tests will show NDSL with RICE performance in (1)different time steps, (2)different splitting forcing, and (3)different initial condition.