Stereographic projection for three‐dimensional global discontinuous Galerkin atmospheric modeling

Abstract

A method to solve the three-dimensional compressible Navier-Stokes equations on the sphere is suggested, based on a stereographic projection with a high-order mapping of the elements from the stereographic space to the sphere. The projection is slightly modified, in order to take into account the domain thickness without introducing any approximation about the aspect ratio (deep-atmosphere). In a discontinuous Galerkin framework, the elements alongside the equator are exactly represented using a nonpolynomial geometry, in order to avoid the numerical issues associated with the seam connecting the two hemispheres. This is an crucial point, necessary to avoid mass loss and spurious deviations of the velocity. The resulting model is validated on idealized three-dimensional atmospheric test cases on the sphere, demonstrating the good convergence properties of the scheme, its mass conservation, and its satisfactory behavior in terms of accuracy and low numerical dissipation. A simulation is performed on a variable resolution unstructured grid, producing accurate results despite a substantial reduction of the number of elements.

A stabilization for three‐dimensional discontinuous Galerkin discretizations applied to nonhydrostatic atmospheric simulations

“…This effect is confirmed by a stability analysis, demonstrating that the reduction of the polynomial degree discards unstable modes from the discrete spatial operator (Section 6.1). This local projection of the gravity term has no effect on mass conservation, which is still ensured up to machine precision [38]. The computational overhead, resulting only from the local projection operator, is below 5% of the total computational time.…”

“…This local projection of the gravity term has no effect on mass conservation, which is still ensured up to machine precision . The computational overhead, resulting only from the local projection operator, is below 5% of the total computational time.…”