Implicit dual time-stepping positivity-preserving entropy-stable schemes for the compressible Navier-Stokes equations
This work addresses the need for stable and accurate numerical methods for high Reynolds number viscous flows in computational fluid dynamics, representing an incremental advancement by extending existing explicit schemes to implicit formulations.
The authors tackled the challenge of simulating compressible Navier-Stokes equations by generalizing explicit high-order schemes to an implicit formulation, achieving provable entropy stability and positivity-preserving properties with unconditional stability in physical time, as demonstrated in numerical tests for supersonic viscous flows with strong shocks.
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.