  Rational exponential integrators (REXI) are a class of numerical methods that
are well suited for the time integration of linear partial differential
equations with imaginary eigenvalues. Since these methods can be parallelized
in time (in addition to the spatial parallelization that is commonly performed)
they are well suited to exploit modern high performance computing systems. In
this paper, we propose a novel REXI scheme that drastically improves accuracy
and efficiency. The chosen approach will also allow us to easily determine how
many terms are required in the approximation in order to obtain accurate
results. We provide comparative numerical simulations for a shallow water
equation that highlight the efficiency of our approach and demonstrate that
REXI schemes can be efficiently implemented on graphic processing units.
