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

TL;DR

This paper explores higher-order exponential Rosenbrock time integration methods for the shallow water equations on the sphere, demonstrating improved efficiency and accuracy over previous methods in weather modeling tests.

Contribution

It introduces and evaluates new exponential Rosenbrock methods with modifications to the phi-function computation, showing enhanced performance for meteorological simulations.

Findings

Methods enable longer time-steps for accurate solutions.

Proposed methods outperform epi3 in efficiency and accuracy.

Effective for highly nonlinear and complex test problems.

Abstract

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.