AI 中文总结
本文将显式谱配点格式扩展为隐式BDF2双时间步格式，用于高雷诺数非定常可压缩黏性流动，该格式保正且熵稳定，数值验证了其效率与精度。
AI 中文摘要
本文对Upperman 2023和Yamaleev 2023提出的用于三维可压缩Navier-Stokes方程的显式、高阶、保正、熵稳定谱配点格式进行了严格扩展，将其发展为时间隐式格式。时间导数项采用二阶隐式向后差分公式（BDF2）离散，该公式适用于求解高雷诺数下随时间变化的黏性流动。每个物理时间步由BDF2离散得到的非线性离散方程组采用双时间步（DTS）技术求解。该BDF2 DTS格式在伪时间上是熵稳定且保正的，在物理时间上具有无条件稳定性。数值结果展示了该保正BDF2 DTS格式相较于其显式对应格式的效率与精度，针对带有强激波和接触间断的超声速流动进行了验证。
英文摘要
This paper presents a rigorous extension of the explicit, high-order, positivity-preserving, and entropy-stable spectral collocation schemes developed in Upperman 2023 and Yamaleev 2023 for the 3D compressible Navier-Stokes equations to a time-implicit formulation. The time derivative terms are discretized by using the second-order implicit backward difference formula (BDF2) that is well suited for solving time-variable viscous flows at high Reynolds numbers. The nonlinear system of discrete equations resulting from the BDF2 discretization at each physical timestep is solved using a dual time-stepping (DTS) technique. The BDF2 DTS scheme is entropy-stable and positivity-preserving in the pseudotime and provides unconditional stability properties in the physical time. Numerical results demonstrate the efficiency and accuracy of the positivity-preserving BDF2 DTS scheme as compared with its explicit counterpart are presented for supersonic flows with strong shock waves and contact discontinuities.
