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

Mohammed Sayyari, Nail K. Yamaleev

TL;DR

The paper develops implicit dual-time-stepping schemes (BDF1 and BDF2 in physical time, explicit in pseudotime) to solve the 3D compressible Navier–Stokes equations with Brenner regularization, achieving positivity-preserving and entropy-stable discretizations at high order. It combines SBP-based spectral collocation with entropy-stable fluxes, analyzes pseudotime-step bounds for density and internal-energy positivity, and introduces flux-limiting to maintain positivity for high-order spatial discretizations. Numerical results across 3D shocks, SBLI, hypersonic cylinder, and Taylor–Green vortex demonstrate accuracy, positivity preservation, entropy stability, and significant speed-ups relative to explicit-time schemes. The work provides a practical framework for robust, high-order simulations of viscous, compressible flows at high Mach and Reynolds numbers, while outlining future work on nonlinear positivity-preserving solvers to further enhance efficiency.

Abstract

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.

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

TL;DR

The paper develops implicit dual-time-stepping schemes (BDF1 and BDF2 in physical time, explicit in pseudotime) to solve the 3D compressible Navier–Stokes equations with Brenner regularization, achieving positivity-preserving and entropy-stable discretizations at high order. It combines SBP-based spectral collocation with entropy-stable fluxes, analyzes pseudotime-step bounds for density and internal-energy positivity, and introduces flux-limiting to maintain positivity for high-order spatial discretizations. Numerical results across 3D shocks, SBLI, hypersonic cylinder, and Taylor–Green vortex demonstrate accuracy, positivity preservation, entropy stability, and significant speed-ups relative to explicit-time schemes. The work provides a practical framework for robust, high-order simulations of viscous, compressible flows at high Mach and Reynolds numbers, while outlining future work on nonlinear positivity-preserving solvers to further enhance efficiency.

Abstract

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.

Paper Structure

This paper contains 25 sections, 1 theorem, 49 equations, 9 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Regularized Navier-Stokes equations
  3. Semi-discrete 1st-order positivity-preserving entropy stable scheme
  4. SBP operators
  5. Baseline 1st-order positivity-preserving entropy stable scheme
  6. Dual time-stepping method
  7. First-order dual time-stepping positivity-preserving entropy stable scheme
  8. Positivity of density
  9. Positivity of internal energy
  10. Dual time-stepping positivity-preserving entropy stable BDF1 scheme with high-order spatial discretization
  11. High-order positivity-violating entropy stable scheme
  12. High-order positivity--preserving entropy stable flux-limiting scheme
  13. Dual time-stepping BDF2 positivity-preserving scheme
  14. Density positivity
  15. Internal energy positivity
  16. ...and 10 more sections

Key Result

Theorem 1

If the $\hat{\bar{f}}^{\rho\pm}_{i_d}$ flux is defined by Eq. (eq:flux) with $\mathscr{D}^{\pm}_{i_d} \geq \mathscr{D}^{\pm}_{{i_d},\min}= \frac{|\hat{\bar{m}}^{\pm}_{i_d}|}{2{\rho}_{i_d,A}^{\pm}}$, then the first-order dual time-stepping scheme given by Eq. (eq:bdf1-update) preserves the positivity

Figures (9)

  • Figure 1: The wall pressure (left panel) and skin friction coefficient computed with the BDF1 dual time-stepping and forward Euler $p=4$ schemes on the $N_\text{elem}=12,012$ grid and the DG $p=6$ method on the $N_\text{elem}=11,041$ grid blanchard2016sbli for the $M_{\infty}=2.15$ SBLI case.
  • Figure 2: Convergence histories obtained with the forward Euler and dual time-stepping BDF1 schemes for the SBLI $M_{\infty}=2.15$ case.
  • Figure 3: Mach number contours computed with the dual time-stepping BDF1 $p=4$ scheme for the $M_{\infty}=6.85$ SBLI case on the $N_\text{elem}=27,990$ grid.
  • Figure 4: The wall pressure (left panel) and skin friction coefficient computed with the dual time-stepping BDF1 and forward Euler $p=4$ schemes on the $N_\text{elem}=27,990$ grid for the $M_{\infty}=6.85$ SBLI case.
  • Figure 5: Mach number (left panel) and vorticity contours computed with the BDF2 dual time-stepping $p=5$ scheme for the hypersonic cylinder flow at $M_{\infty}=17.605$ on the $55,216$ element grid.
  • ...and 4 more figures

Theorems & Definitions (5)

  • Remark 1
  • Theorem 1
  • proof
  • Remark 2
  • Remark 3