  A novel numerical approach to solving the shallow-water equations on the
sphere using high-order numerical discretizations in both space and time is
proposed. A space-time tensor formalism is used to express the equations of
motion covariantly and to describe the geometry of the rotated cubed-sphere
grid. The spatial discretization is done with the direct flux reconstruction
method, which is an alternative formulation to the discontinuous Galerkin
approach. The equations of motion are solved in differential form and the
resulting discretization is free from quadrature rules. It is well known that
the time step of traditional explicit methods is limited by the phase velocity
of the fastest waves. Exponential integration is employed to enable
integrations with significantly larger time step sizes and improve the
efficiency of the overall time integration. New multistep-type exponential
propagation iterative methods of orders 4, 5 and 6 are constructed and applied
to integrate the shallow-water equations in time. These new schemes enable time
integration with high-order accuracy but without significant increases in
computational time compared to low-order methods. The exponential matrix
functions-vector products used in the exponential schemes are approximated
using the complex-step approximation of the Jacobian in the Krylov-based KIOPS
(Krylov with incomplete orthogonalization procedure solver) algorithm.
Performance of the new numerical methods is evaluated using a set of standard
benchmark tests.
