Spectral Difference method with a posteriori limiting: II- Application to low Mach number flows
This addresses numerical diffusion issues in astrophysical fluid solvers for stellar convection, though it is incremental as it builds on existing Spectral Difference methods with specific adaptations.
The paper tackled the challenges of simulating low-Mach number flows and small perturbations in stellar convection by applying a high-order Spectral Difference method with a posteriori limiting, showing that a fourth-order scheme optimally captures turbulent kinetic energy evolution.
Stellar convection poses two main gargantuan challenges for astrophysical fluid solvers: low-Mach number flows and minuscule perturbations over steeply stratified hydrostatic equilibria. Most methods exhibit excessive numerical diffusion and are unable to capture the correct solution due to large truncation errors. In this paper, we analyze the performance of the Spectral Difference (SD) method under these extreme conditions using an arbitrarily high-order shock capturing scheme with a posteriori limiting. We include both a modification to the HLLC Riemann solver adapted to low Mach number flows (L-HLLC) and a well-balanced scheme to properly evolve perturbations over steep equilibrium solutions. We evaluate the performance of our method using a series of test tailored specifically for stellar convection. We observe that our high-order SD method is capable of dealing with very subsonic flows without necessarily using the modified Riemann solver. We find however that the well-balanced framework is unavoidable if one wants to capture accurately small amplitude convective and acoustic modes. Analyzing the temporal and spatial evolution of the turbulent kinetic energy, we show that our fourth-order SD scheme seems to emerge as an optimal variant to solve this difficult numerical problem.