High-order Godunov methods for gas dynamics have become a standard tool for simulating different classes of astrophysical flows. Their accuracy is mostly determined by the spatial interpolant used to reconstruct the pair of Riemann states at cell interfaces and by the Riemann solver that computes the interface fluxes. In most Godunov-type methods, these two steps can be treated independently, so that many different schemes can in principle be built from the same numerical framework. In this work, we use our fully compressible Seven-League Hydro (SLH) code to test the accuracy of six reconstruction methods and three approximate Riemann solvers on two- and three-dimensional (2D and 3D) problems involving subsonic flows only. We consider Mach numbers in the range from $10^{-3}$ to $10^{-1}$ in a well-posed, 2D, Kelvin--Helmholtz instability problem and a 3D turbulent convection zone that excites internal gravity waves in an overlying stable layer. We find that (i) there is a spread of almost four orders of magnitude in computational cost per fixed accuracy between the methods tested in this study, with the most performant method being a combination of a "low-dissipation" Riemann solver and a sextic reconstruction scheme, (ii) the low-dissipation solver always outperforms conventional Riemann solvers on a fixed grid when the reconstruction scheme is kept the same, (iii) in simulations of turbulent flows, increasing the order of spatial reconstruction reduces the characteristic dissipation length scale achieved on a given grid even if the overall scheme is only second order accurate, (iv) reconstruction methods based on slope-limiting techniques tend to generate artificial, high-frequency acoustic waves during the evolution of the flow, (v) unlimited reconstruction methods introduce oscillations in the thermal stratification near the convective boundary, where the entropy gradient is steep.

翻译：高阶Godunov方法用于气体动力学已成为模拟各类天体物理流动的标准工具。其精度主要取决于用于重构单元界面处黎曼状态对的空间插值格式，以及计算界面通量的黎曼求解器。在大多数Godunov型方法中，这两个步骤可独立处理，因此原则上可从同一数值框架构建多种不同格式。本研究利用完全可压缩的Seven-League Hydro（SLH）代码，测试了六种重构方法和三种近似黎曼求解器在仅涉及亚声速流动的二维和三维问题中的精度。我们考虑马赫数范围为$10^{-3}$到$10^{-1}$，在具有良好适定性的二维Kelvin–Helmholtz不稳定性问题，以及激发覆盖稳定层中内重力波的三维湍流对流区中展开实验。研究发现：（i）在本研究测试的方法中，固定精度下的计算成本差异近四个数量级，性能最优的组合为"低耗散"黎曼求解器与六次重构格式；（ii）在相同重构方案下，固定网格上低耗散求解器始终优于传统黎曼求解器；（iii）在湍流流动模拟中，即使整体格式仅具有二阶精度，提高空间重构阶数可减小给定网格上的特征耗散长度尺度；（iv）基于斜率限制技术的重构方法在流动演化过程中易产生人为的高频声波；（v）无限制重构方法会在对流边界附近热分层陡峭处（熵梯度剧烈区域）引入振荡。