A fully-implicit method to simulate convection in spherical shells*

We present a fully-implicit method to simulate Boussinesq convection in spherical shells. This method combines the discrete exterior calculus and second-order central finite difference methods. The governing equations are formulated as a non-linear function and are solved fully implicitly using the backward Euler time integration scheme, where the Jacobian is hand-coded and utilized in the integration. We use a triangulated spherical surface mesh that is distributed across the MPI ranks, and a Chebyshev grid along the radial direction for higher resolution of the boundary layers. We apply the method of manufactured solutions (MMS) and observe second-order spatial convergence. The method is verified through various test cases, including the determination of critical Rayleigh numbers for spherical shells of varying aspect ratios, simulation of spherical spiral rolls in moderately thin shells, and Nusselt-Rayleigh number scaling. We also present strong and weak scaling tests conducted on the Shaheen-III supercomputer.