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An optimization based limiter for enforcing positivity in a semi-implicit discontinuous Galerkin scheme for compressible Navier-Stokes equations

Chen Liu, Gregery T. Buzzard, Xiangxiong Zhang

Abstract

We consider an optimization based limiter for enforcing positivity of internal energy in a semi-implicit scheme for solving gas dynamics equations. With Strang splitting, the compressible Navier-Stokes system is splitted into the compressible Euler equations, solved by the positivity-preserving Runge-Kutta discontinuous Galerkin (DG) method, and the parabolic subproblem, solved by Crank-Nicolson method with interior penalty DG method. Such a scheme is at most second order accurate in time, high order accurate in space, conservative, and preserves positivity of density. To further enforce the positivity of internal energy, we impose an optimization based limiter for the total energy variable to post process DG polynomial cell averages. The optimization based limiter can be efficiently implemented by the popular first order convex optimization algorithms such as the Douglas-Rachford splitting method if using the optimal algorithm parameters. Numerical tests suggest that the DG method with $\mathbb{Q}^k$ basis and the optimization-based limiter is robust for demanding low pressure problems such as high speed flows.

An optimization based limiter for enforcing positivity in a semi-implicit discontinuous Galerkin scheme for compressible Navier-Stokes equations

Abstract

We consider an optimization based limiter for enforcing positivity of internal energy in a semi-implicit scheme for solving gas dynamics equations. With Strang splitting, the compressible Navier-Stokes system is splitted into the compressible Euler equations, solved by the positivity-preserving Runge-Kutta discontinuous Galerkin (DG) method, and the parabolic subproblem, solved by Crank-Nicolson method with interior penalty DG method. Such a scheme is at most second order accurate in time, high order accurate in space, conservative, and preserves positivity of density. To further enforce the positivity of internal energy, we impose an optimization based limiter for the total energy variable to post process DG polynomial cell averages. The optimization based limiter can be efficiently implemented by the popular first order convex optimization algorithms such as the Douglas-Rachford splitting method if using the optimal algorithm parameters. Numerical tests suggest that the DG method with basis and the optimization-based limiter is robust for demanding low pressure problems such as high speed flows.
Paper Structure (32 sections, 5 theorems, 59 equations, 10 figures, 4 tables)

This paper contains 32 sections, 5 theorems, 59 equations, 10 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Motivation and objective
  3. Existing positivity-preserving schemes for compressible NS equations
  4. A constraint optimization approach for enforcing positivity and global conservation
  5. Efficient implementation of the constraint optimization defined postprocessing
  6. The main result and organization of this paper
  7. Numerical scheme
  8. Time discretization
  9. Preliminary aspects of space discretization
  10. Mesh, approximation spaces, and quadratures
  11. Hyperbolic subproblem
  12. Parabolic subproblem
  13. The simple positivity-preserving limiter
  14. The fully discrete scheme
  15. Global conservation of the fully discrete scheme
  16. ...and 17 more sections

Key Result

Theorem 1

Assume ${\boldsymbol{U}}_h^\mathrm{P}({\boldsymbol{x}}_q)$ belongs to the set of admissible states for all ${\boldsymbol{x}}_q\in S_h$, then the fully discrete scheme conserves density, momentum, and total energy. We have

Figures (10)

  • Figure 1: An illustration of the quadratures used in the $\mathds{Q}^4$ scheme. From left to right: the quadrature points for face integrals in ($\mathrm{H}$), volume integrals in ($\mathrm{H}$), face integrals in ($\mathrm{P}$), volume integrals in ($\mathrm{P}$), and the quadrature points for weak positivity. The black points are used only in defining the positivity-preserving limiter, and they are not used in calculating any numerical integration.
  • Figure 2: DG with $\mathds{Q}^2$ basis for 2D Sedov blast wave test. The middle figure is the zoom view of the left figure: the shock is marked black; the negative cells are highlighted by the red marks; by the definition \ref{['definition-T']}, $T$ does not include cells in the gray region in which the exact solution is supposed to be a constant. Right: the actual convergence rate of the Douglas--Rachford splitting algorithm \ref{['eq:DR_algorithm2']} with nearly optimal parameters \ref{['DR_parameter']} for solving \ref{['total-energy-opt-2']} for the 2D Sedov problem (at one particular time step for the left figure) matches well the predicated rate from analysis (asymptotic linear convergence from analysis using the estimated principle angle $\hat{\theta}=\cos^{-1}\sqrt{\frac{\hat{r}}{|T|}}$), see liu2023simple for more details on such a provable convergence rate.
  • Figure 3: Lax shock tube. The density field snapshots at time $T = 1.3$ are displayed in the mountain view.
  • Figure 4: Double rarefaction. The density field snapshots at time $T = 0.6$ are displayed in the mountain view.
  • Figure 5: Sedov blast wave. The snapshots of density profile are taken at $T=1$. Plot of density: $50$ exponentially distributed contour lines of density from $0.001$ to $6$.
  • ...and 5 more figures

Theorems & Definitions (15)

  • Remark 1
  • Theorem 1
  • proof
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Lemma 1
  • proof
  • Lemma 2
  • ...and 5 more