A Conservative Semi-Lagrangian Discontinuous Galerkin Scheme on the Cubed Sphere

Abstract

The discontinuous Galerkin (DG) methods designed for hyperbolic problems arising from a wide range of applications are known to enjoy many computational advantages. DG methods coupled with strong-stability-preserving explicit Runge–Kutta discontinuous Galerkin (RKDG) time discretizations provide a robust numerical approach suitable for geoscience applications including atmospheric modeling. However, a major drawback of the RKDG method is its stringent Courant–Friedrichs–Lewy (CFL) stability restriction associated with explicit time stepping. To address this issue, the authors adopt a dimension-splitting approach where a semi-Lagrangian (SL) time-stepping strategy is combined with the DG method. The resulting SLDG scheme employs a sequence of 1D operations for solving multidimensional transport equations. The SLDG scheme is inherently conservative and has the option to incorporate a local positivity-preserving filter for tracers. A novel feature of the SLDG algorithm is that it can be used for multitracer transport for global models employing spectral-element grids, without using an additional finite-volume grid system. The quality of the proposed method is demonstrated via benchmark tests on Cartesian and cubed-sphere geometry, which employs nonorthogonal, curvilinear coordinates.

A Positive‐Definite, WENO‐Limited, High‐Order Finite Volume Solver for 2‐D Transport on the Cubed Sphere Using an ADER Time Discretization

“…Notice how large the errors are for the Strang splitting. This sort of error at cube corners for Strang splitting was also seen in Guo et al (). These errors can be reduced with a smaller time step, but that jump always seems to be there.…”

“…Typically in 2‐D, one would only do an x and a y sweep with dimensional splitting. However, the cubed sphere requires three sweeps across the domain, and the details of this are specified in Guo et al (). The directions ξ , η , and ζ are indicative of three global closed directed paths over the surface of the sphere.…”

“…Second, one cannot efficiently implement a truly multidimensional Lagrangian operator for the dynamics . There are dimensionally split characteristics‐based dynamics implementations, but it has been shown that true dimensional splitting on the cubed sphere gives very large errors in the norm when large time steps are used (Guo et al, ; Norman et al, ). There are fully multidimensional characteristic‐based implementations (Lukáčová‐Medvid'ová et al, , ), but they are usually rendered semidiscrete, and in a fully discrete context, they would require integrating over skewed Mach cones in 2‐D and Mach ellipsoids in 3‐D—operations that are prohibitively expensive.…”

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

“…Substituting the discretized scalar field (5) and test functions into (2), and replacing integrals by the Gaussian-Lobatto quadratures, converts the partial differential equation into a set of ordinary differential equations in time, which may be written abstractly as…”

Comparison of dimensionally split and multi‐dimensional atmospheric transport schemes for long time steps

“…They create ghost cells outside each cube panel boundary and achieve accuracy between second and fourth order. Guo et al (2014) report error growth around cube edges using Strang splitting.…”