Utilizing a mapping $(\xi, \eta) \in \mathbb{R}^2 \rightarrow (x, y, z) \in \mathbb{R}^3$ and specifically tailored tensor-product Legendre–Gauss–Lobatto basis functions [1,2], discontinuous Galerkin (DG) simulations can be performed on a curved surface. In particular, we are interested in running DG simulations on a spherical shell.

To achieve this, we create a two-dimensional cubed-sphere mesh for tessellating the sphere’s surface. This mesh proves advantageous as it avoids singularities at the poles present in latitude-longitude grids, while still facilitating a highly regular tessellation of the simulation domain through the use of quadrilaterals.

Within this mesh, we address the linear advection equations incorporating position-dependent advection velocity:
\[ \frac{\partial \rho}{\partial t} + \nabla \cdot \left( \vec{v} (x, y, z) \rho \right) = 0. \]
Additionally, a solid-body rotation velocity tangential to the spherical surface is imposed.

The presented video showcases the advection of a Gaussian pulse across the globe using Trixi.jl (https://github.com/trixi-framework/Trixi.jl) with the two-dimensional p4est solver. The simulation also incorporates a custom implementation of the three-dimensional linear advection equations.

References

[1] Song, C. & Wolf, J. P. (1999). The scaled boundary finite element method—alias consistent infinitesimal finite element cell method—for diffusion. International Journal for Numerical Methods in Engineering, 45(10), 1403-1431.
[2] Giraldo, F. X. (2001). A spectral element shallow water model on spherical geodesic grids. International Journal for Numerical Methods in Fluids, 35(8), 869-901.