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

TL;DR

The paper tackles the challenge of simulating very low Mach number astrophysical flows with small perturbations over hydrostatic equilibria by deploying an arbitrarily high-order Spectral Difference method equipped with a posteriori limiting and a well-balanced formulation. It combines a modified low-Mach Riemann solver (L-HLLC), flux blending, nonturbulent perturbation evolution, and robust limiting to handle both smooth subsonic dynamics and sharp interfaces. Key findings show exponential convergence for smooth, subsonic flows and substantial advantages of high-order SD in capturing tiny perturbations and developing small-scale structures, with the 4th-order SD (SD4BL) offering the best cost-to-accuracy ratio in many tests. The work demonstrates that explicit time integration remains viable without mandatory low-Mach corrections, highlights the necessity of well-balancing for stellar convection problems, and provides insights into the trade-offs between increasing polynomial order and refining spatial resolution for subsonic astrophysical simulations.

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.

Figures (11)

• Figure 1: Flux blending: On the left in red the troubled sub-cells. On the right the values for the blending coefficient $\theta$
• Figure 2: Gresho vortex test: Color maps for the control volume average of $\mathcal{M} = v_{\phi}/c_s$. Results at $t = 1$ for three values of the Mach number ($\mathcal{M}_{\max}=10^{-1},10^{-2}$ and $10^{-3}$) and a background velocity $v_0=5$. On the first row the results for FV2 (with HLLC), on the second row FV2L (with L-HLLC), and and on the third row SD3 ($p=2$). All of them making use of $96^2$ DOF.
• Figure 3: Gresho vortex test: 1-dimensional slices of the control volume average of $\mathcal{M} = v_{\phi}/c_s$ at $t=1$, corresponding to five laps over the domain ($v_0=5$).
• Figure 4: Rayleigh Taylor instability test. Colormaps for the density at $t=1.95$ for 2, 4 and 8 order of the numerical approximation for $48\times192$ DOF. On the first row the results for $P_0=1$, on the second row the results for $P_0=10$ and on the third row the results for $P_0=100$.
• Figure 5: Rayleigh Taylor instability test. Iso-contours for the density at $t=1$ for 2, 4 and 8 order of the numerical approximation for $P_0=100$. On the first row the results with $96\times384$ DOF, on the second row the results with $192\times768$ DOF.