Further development of efficient and accurate time integration schemes for meteorological models
For meteorological modelers, the paper offers more efficient time integration schemes for stiff nonlinear models, though the improvements are incremental over existing methods.
The paper investigates higher-order exponential Rosenbrock methods for the shallow water equations on the sphere, proposing modifications to the phipm_IOM2 algorithm for efficiency. The proposed methods achieve accurate solutions at much longer time-steps than the previous epi3 method, proving more efficient as error tolerance decreases or nonlinearity increases.
In this paper, we investigate the use of higher-order exponential Rosenbrock time integration methods on 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 on this model. Based on their multi-stage structure, we propose a set of modifications to the phipm_IOM2 algorithm for efficiently calculating the matrix phi-functions. 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.