Verification of a hybrid discrete exterior calculus and finite difference solver for density stratified convection in spherical shells

Stratification and buoyancy contribute to complex flow phenomena in solar thermal convection. Besides that, the sun’s spherical geometry makes the numerical simulation significantly more challenging. Discrete exterior calculus (DEC) has recently gained popularity in the simulation and modeling communities as a powerful numerical method with coordinate independence and for simulating flows on curved surfaces. However, there is a dearth of DEC solvers for density-stratified thermal convection in a spherical shell.

The present work utilizes a hybrid discrete exterior calculus and finite difference (DEC-FD) solver to investigate density-stratified convection in spherical shells. The discrete exterior calculus (DEC) is employed here to compute the spherical surface flow, and the finite difference (FD) is used to calculate the radial flow, following Mantravadi et al. (arXiv preprint arXiv:2210.00861) for Boussinesq convection. The discretization of governing equations is carried out through the DEC and FD methods. Our in-house developed (DEC-FD) solver is verified using the method of manufactured solution (MMS), a conventional and classical method to test a solver. The MMS approach is well-accepted for quantifying numerical methods within complex physical systems. This methodology ensures a rigorous and convincing verification of numerical accuracy through systematic grid convergence tests. By using the MMS procedure, this work verifies the order of accuracy of DEC-FD discretized equations, which is in good agreement with theoretical order. Furthermore, simulations at different density ratios demonstrate the hybrid DEC-FD solver assessment for stratified convection in a basally heated spherical shell.

Acknowledgements

This publication is based upon work supported by the King Abdullah University of Science and Technology (KAUST) Office of Sponsored Research (OSR) under Award URF/1/4342-01. For computer time, this research used the Cray XC40, Shaheen II, of the Supercomputing Laboratory at King Abdullah University of Science and Technology (KAUST) in Thuwal, Saudi Arabia.