Implicit dual time-stepping positivity-preserving entropy-stable schemes for the compressible Navier-Stokes equations

Abstract

We generalize the explicit high-order positivity-preserving entropy stable spectral collocation schemes developed in Upperman 2023 and Yamaleev 2023 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 time step 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 the 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.