HLI, A Direct Method Suitable for Partial and Fully Implicit Time Integration of Primitive Equation Meteorological Models

HLI, the homogeneously linearized implicit method is a direct implicit time integration method for nonlinear prognostic equations. For homogeneous linear test problems it is identical to the known implicit method, when applied to the full model area. As most theoretical results concerning the implicit method use linear homogeneous test problems, these apply immediately to HLI. The implicit method, when applied to the whole model area, is called the fully implicit method. In order to reduce the numerical expense for large integration areas, a partial implicit method is defined by posing artificial numerical boundary conditions, which result in a smaller set of equations, corresponding to the part of the integration area chosen. The application of either full or partial methods to nonlinear and inhomogeneous problems is achieved by defining for each grid point the local homogeneously linearized problem. Partial schemes are convenient when using multiprocessor computers, as only local and neighbouring data have to be used at each grid point. Only grid point discretisations of finite order are admitted. The proposed method would not be suitable for the spectral method. Computational examples are presented to show the suitability of this method for inhomogeneous situations and the ease of the treatment of internal boundary conditions. A semi-implicit treatment was also tested, which treats only the terms responsible for the fast waves implicit. This requires only the Fourier transformation of one field and in the generalised Fourier back transformation only one linear equation has to be solved, rather that a system of equations. In this paper only the trapezoidal implicit method was used. HLI could, however, be applied in the same way with other implicit methods.