Solving Helmholtz equations of the GRAPES global nonhydrostatic model based on the PETSc.

Starting from the Eulerian operator, the forecasting equations with 2order time and space differential remainder are derived in light of the Taylor series expansion.The quasiLagrangian integration scheme with an algorithm of numerical analysis the-bicubic surface interpolation-for a new meteorological numerical model is then suggested. It describes atmospheric motions just as nonlinear cubic waves, with fitting different bicubic surfaces to physical scalar or vector variable fields in the model atmosphere, i.e. the pressure, temperature, humidity, wind and divergence as well as the generalized Newtonian force acting to unit air mass on the rotating earth (acceleration). With the procedure of “fitting bicubic surface-time step integration-fitting bicubic surface……” through oneto one correspondecnce with the Hermite bicubic patches to parameterize the latitudelongitudinal meshes in light of rectangular topologies so as to get the secondorder derivative, a new model’s dynamic core comes true. Because of that the bicubic surface has two numerical analysis laws of the convergence and the optimality of the second-order derivative, it is necessary and sufficient reason mathematically to select the bicubic surface interpolation for calculating the upstream point of a quasi-Lagrangian air parcel. It covers the slope, curvature and torsion of every nonlinear variable in the atmospheric motions, that is, one Hermite bicubic patch is mathematically equivalent to one secondary derivative (bicubic) variable “mesh” in the model atmosphere, and the former is “convergent”to the latter-with “optimality”. The quasiLagrangian integration of calculating the upstream point with interpolation on the Hermite bicubic patch is also equivalent to the Eulerian integration with the unitive CFL computational stability criterion. It’s easy to fit a holoscopic variable field to a global bicubic surface, and make some reasonable local area or single point smoothing, according to some conditions of its slope, curvature or torsion, which must conform to the physical constraints such as mass conservation for pressure smooth, energy conservation for temperature smooth, and momentum conservation for wind smooth. Obviously it’s easy to put a new “patch” of bicubic surface on the smoothed place.