  The paper develops high order accurate Runge-Kutta discontinuous local
evolution Galerkin (RKDLEG) methods on the cubed-sphere grid for the shallow
water equations (SWEs). Instead of using the dimensional splitting method or
solving one-dimensional Riemann problem in the direction normal to the cell
interface, the RKDLEG methods are built on genuinely multi-dimensional
approximate local evolution operator of the locally linearized SWEs on a sphere
by considering all bicharacteristic directions. Several numerical experiments
are conducted to demonstrate the accuracy and performance of our RKDLEG
methods, in comparison to the Runge-Kutta discontinuous Galerkin method with
Godunov's flux etc.
