An efficient WENO limiter for discontinuous Galerkin transport scheme on the cubed sphere

The discontinuous Galerkin (DG) transport scheme is becoming increasingly popular in the atmospheric modeling due to its distinguished features, such as high-order accuracy and high-parallel efficiency. Despite the great advantages, DG schemes may produce unphysical oscillations in approximating transport equations with discontinuous solution structures including strong shocks or sharp gradients. Nonlinear limiters need to be applied to suppress the undesirable oscillations and enhance the numerical stability. It is usually very difficult to design limiters to achieve both high-order accuracy and non-oscillatory properties and even more challenging for the cubed-sphere geometry. In this paper, a simple and efficient limiter based on the weighted essentially non-oscillatory (WENO) methodology is incorporated in the DG transport framework on the cubed sphere. The uniform high-order accuracy of the resulting scheme is maintained because of the high-order nature of WENO procedures. Unlike the classic WENO limiter, for which the wide halo region may significantly impede parallel efficiency, the simple limiter requires only the information from the nearest neighboring elements without degrading the inherent high-parallel efficiency of the DG scheme. A bound-preserving filter can be further coupled in the scheme that guarantees the highly desirable positivity-preserving property for the numerical solution. The resulting scheme is high-order accurate, non-oscillatory, and positivity preserving for solving transport equations based on the cubed-sphere geometry. Extensive numerical results for several benchmark spherical transport problems are provided to demonstrate good results, both in accuracy and in non-oscillatory performance.