High-Order Finite-Volume Transport on the Cubed Sphere: Comparison between 1D and 2D Reconstruction Schemes

Abstract

Abstract This paper presents two finite-volume (FV) schemes for solving linear transport problems on the cubed-sphere grid system. The schemes are based on the central-upwind finite-volume (CUFV) method, which is a class of Godunov-type method for solving hyperbolic conservation laws, and combines the attractive features of the classical upwind and central FV methods. One of the CUFV schemes is based on a dimension-by-dimension approach and employs a fifth-order one-dimensional (1D) Weighted Essentially Nonoscillatory (WENO5) reconstruction method. The other scheme employs a fully two-dimensional (2D) fourth-order accurate reconstruction method. The cubed-sphere grid system imposes several computational challenges due to its patched-domain topology and nonorthogonal curvilinear grid structure. A high-order 1D interpolation procedure combining cubic and quadratic interpolations is developed for the FV schemes to handle the discontinuous edges of the cubed-sphere grid. The WENO5 scheme is compared against the fourth-order Kurganov–Levy (KL) scheme formulated in the CUFV framework. The performance of the schemes is compared using several benchmark problems such as the solid-body rotation and deformational-flow tests, and empirical convergence rates are reported. In addition, a bound-preserving filter combined with an optional positivity-preserving filter is tested for nonsmooth problems. The filtering techniques considered are local, inexpensive, and effective. A fourth-order strong stability preserving explicit Runge–Kutta time-stepping scheme is used for integration. The results show that schemes are competitive to other published FV schemes in the same category.