Before revision:  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.
After revision:  New multistep 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 is applied to the Krylov-based KIOPS (Krylov with incomplete orthogonalization procedure solver) algorithm for computing matrix-vector products with \varphi -functions.