Spectral Difference method with a posteriori limiting: II- Application to low Mach number flows

TL;DR

This paper evaluates a high-order spectral difference method with a posteriori limiting for low Mach number stellar convection flows, emphasizing the importance of a well-balanced scheme for accuracy.

Contribution

It introduces a modified Riemann solver and a well-balanced scheme within the spectral difference method to improve simulation of low Mach number stellar convection.

Findings

High-order SD method handles very subsonic flows without the modified Riemann solver.

Well-balanced framework is essential for accurately capturing small perturbations.

Fourth-order SD scheme is identified as an optimal approach for these conditions.

Abstract

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.