A Fast and Accurate Numerical Scheme for the Primitive Equations of the Atmosphere

Abstract

We present a fast and accurate numerical scheme for the approximation of the primitive equations of the atmosphere. The temporal variable is discretized by using a special semi-implicit scheme which requires only to solve a Helmholtz equation and a nonlocal Stokes problem at each time step; the spatial variables are discretized by a spectral-Galerkin procedure with the horizontal components of vectorial spherical harmonics for the horizontal variables and Legendre or Chebyshev polynomials for the vertical variable. The new scheme has two distinct features: (i) it is unconditionally stable given fixed physical parameters, and (ii) the Helmholtz equation and the nonlocal Stokes problem which need to be solved at each time step can be decomposed into a sequence of one-dimensional equations (in the vertical variable) which can be solved by a spectral-Galerkin method with optimal computational complexity.