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.<\/p>\n
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.<\/p>\n
Within this mesh, we address the linear advection equations incorporating position-dependent advection velocity:
\n\\[ \\frac{\\partial \\rho}{\\partial t} + \\nabla \\cdot \\left( \\vec{v} (x, y, z) \\rho \\right) = 0. \\]
\nAdditionally, a solid-body rotation velocity tangential to the spherical surface is imposed.<\/p>\n
The presented video showcases the advection of a Gaussian pulse across the globe using Trixi.jl (https:\/\/github.com\/trixi-framework\/Trixi.jl<\/a>) with the two-dimensional p4est solver. The simulation also incorporates a custom implementation of the three-dimensional linear advection equations.<\/p>\n