  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.
