For the equation set of the non-hydrostatic icosahedral global climate model being developed at Frontier Research System for Global Change, we have devised a conservative scheme of the compressible non-hydrostatic equations. The scheme is based on the flux form equations of total density, three components of momentum, total energy, and densities of water substances. Time-splitting is used with the leap-frog or 2nd order Runge-Kutta schemes for the large time step integration, and sound and gravity waves are treated implicitly in the vertical direction and explicitly in the horizontal directions for the small time step integration. Energy correction is introduced to ensure the conservation of total energy at every small time integration. The hydrological process including the warm rain cloud processes is also incorporated with the flux form equations. The scheme is tested with the non-hydrostatic core of the new global model.

Besides the conservations, two improvements have been devised in the model. First, we use more accurate formulas of the thermodynamics of the moist atmosphere by taking account of the effects of specific heats of water substances and the dependence of latent heat on temperature. These effects are generally neglected in most numerical models. Second, we introduced a conservative semi-Lagrangian method for the transportations due to rain.

We have performed experiments of a squall-line and direct calculation of a radiative-convective equilibrium to confirm the conservations of mass and energy. We found that, if the accurate moist thermodynamics are used, the total rain is reduced more than 10\% in the squall-line experiment in comparison to the case when the usual simplified thermodynamics are used. We also found that the change in energy due to transportation of rain are very large and cannot be negligible in the flux-form formulation, while that in momentum could be

negligible.