  High-order numerical methods are applied to the shallow-water equations on
the sphere. 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 speed of
the fastest waves. Exponential time integration schemes remove this stability
restriction and allow larger time steps. New multistep exponential propagation
iterative methods of orders 4, 5 and 6 are introduced. The complex-step
approximation of the Jacobian is applied to the Krylov-based KIOPS (Krylov with
incomplete orthogonalization procedure solver) algorithm for computing
matrix-vector products with {\varphi}-functions. Results are evaluated using
standard benchmarks.
