An economical second-order advection scheme for numerical weather prediction

A very simple second-order Eulerian scheme for the advection equation, base d on a forward (backward) time integration on even (odd) grid points, is st udied. The proposed scheme is similar, but not equivalent, to the so-called 'hopscotch method', developed in the 1960s and early 1970s for the advecti on-diffusion equation, and is stable up to Courant number 2.0. It is shown that, in the case of the advection equation, the proposed scheme has the sa me advantages yielded by the forward-backward scheme in the case of the sha llow-water equations; in particular, it is equivalent to the application of centred time and space differencing on the Eliassen grid. The new scheme, unlike the classical leapfrog scheme, can be coupled to the forward-backwar d integration of the gravity-wave problem in primitive-equation models. With the aid of the proposed scheme, an explicit version of the atmospheric , Bologna Limited-Area Model (developed in recent years at the FISBAT Insti tute of the National Council of Research of Italy) is devised, and a compar ison with the semi-implicit version of the same model is performed. The exp licit version runs with a double time step, achieving the same accuracy wit h significantly less computer time and storage. It is suggested that the ne w time scheme is particularly suitable for numerical weather prediction on massively parallel computing machines based on SIMD (Single Instruction Mul tiple Data) and distributed memory architecture, which strongly penalize no n-local algorithms.