We generalize the explicit high-order positivity-preserving entropy-stable spectral collocation schemes developed in [30, 34] for the three-dimensional (3D) compressible Navier Stokes equations to a time implicit formulation. The time derivative terms are discretized by using the first- and second-order implicit backward difference formulas (BDF1 and BDF2) that are well suited for solving steady-state and time-dependent viscous flows at high Reynolds numbers, respectively. The nonlinear system of discrete equations at each physical timestep is solved by using a dual time-stepping technique. The proposed scheme is provably entropy-stable and positivity-preserving and provides unconditional stability properties in the physical time. Numerical results demonstrating accuracy and positivity-preserving properties of the new dual time-stepping scheme are presented for supersonic viscous flows with strong shock waves and contact discontinuities.

翻译：本文将文献[30, 34]中针对三维可压缩Navier-Stokes方程发展的显式高阶保正熵稳定谱配置格式推广至时间隐式形式。时间导数项采用一阶和二阶隐式后向差分公式（BDF1与BDF2）进行离散，二者分别适用于求解高雷诺数下的定常与非定常粘性流动。每个物理时间步的离散方程非线性系统通过双时间步长技术求解。所提格式可证明具有熵稳定性与保正性，并在物理时间上具备无条件稳定性。通过含强激波与接触间断的超音速粘性流动算例，展示了新双时间步长格式的精度与保正特性。