High-order numerical solutions to the shallow-water equations on the rotated cubed-sphere grid

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.