A factored implicit scheme for numerical weather prediction

Abstract

AbstractThe nonlinear partial differential equations of atmospheric dynamics govern motion on two time scales, a fast one and a slow one. Only the slow‐scale motions are relevant in predicting the evolution of large weather patterns. Implicit numerical methods are therefore attractive for weather prediction, since they permit a large time step chosen to resolve only the slow motions.To develop an implicit method which is efficient for problems in more than one spatial dimension, one must approximate the problem by smaller, usually one‐dimensional problems. A popular way to do so is to approximately factor the multidimensional implicit operator into one‐dimensional operators. The factorization error incurred in such methods, however, is often unacceptably large for problems with multiple time scales.We propose a new factorization method for numerical weather prediction which is based on factoring separately the fast and slow parts of the implicit operator. We show analytically that the new method has small factorization error, which is comparable to other discretization errors of the overall scheme. The analysis is based on properties of the shallow water equations, a simple two‐dimensional version of the fully three‐dimensional equations of atmospheric dynamics.