Further development of efficient and accurate time integration schemes for meteorological models

In this paper, we investigate the use of higher-order exponential Rosenbrock time integration methods for the shallow water equations on the sphere. This stiff, nonlinear model provides a 'testing ground' for accurate and stable time integration methods in weather modeling, serving as the focus for exploration of novel methods for many years. We therefore identify a candidate set of three recent exponential Rosenbrock methods of orders four and five (exprb42, pexprb43 and exprb53) for use in this model. Based on their multi-stage structure, we propose a set of modifications to the phipm/IOM2 algorithm for efficiently calculating the matrix functions φ_k. We then investigate the performance of these methods on a suite of four challenging test problems, comparing them against the epi3 method investigated previously in [1,2] on these problems. In all cases, the proposed methods enable accurate solutions at much longer time-steps than epi3, proving considerably more efficient as either the desired solution error decreases, or as the test problem nonlinearity increases.