BDF Dual Time-Stepping Positivity-Preserving Entropy Stable Spectral Collocation Schemes for 3-D Navier-Stokes Equations

Abstract

We extend the explicit high-order positivity-preserving entropy stable spectral collocation schemes developed in [J. Upperman & N. K. Yamaleev, J. Sci. Comp., Vols. 94(1) and 95(1), 2023] for the 3-D compressible Navier-Stokes equations to a time-implicit formulation. The time derivative is discretized by using the implicit 1st- and 2nd-order backward difference (BDF) schemes that are well suited for solving compressible viscous flows at high Mach and Reynolds numbers considered in this study. To solve the nonlinear system of equations arising from the BDF discretization at each time step, a dual time-stepping technique with the explicit 1st-order forward Euler approximation in the pseudotime is used. This dual time-stepping strategy allows us to combine unconditional stability properties of the implicit time integrator with the positivity and entropy stability properties of the baseline explicit spectral collocation scheme in the pseudotime. We present numerical results that demonstrate both the efficiency and positivity-preserving properties of the proposed dual time-stepping schemes for viscous flows with strong shock waves and contact discontinuities.

Mohammed Sayyari

Postdoctoral Research Associate, Adjunct Assistant Professor

Mohammed is a systems thinker. His Mathematical interests are in Numerical Analysis and Modeling of Evolutionary PDEs. His Scientific interests are in the advancement of accurate, efficient, economic and enviromentally friendly technologies.