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Monolithic convex limiting and implicit pseudo-time stepping for calculating steady-state solutions of the Euler equations

Paul Moujaes, Dmitri Kuzmin

Abstract

In this work, we use the monolithic convex limiting (MCL) methodology to enforce relevant inequality constraints in implicit finite element discretizations of the compressible Euler equations. In this context, preservation of invariant domains follows from positivity preservation for intermediate states of the density and internal energy. To avoid spurious oscillations, we additionally impose local maximum principles on intermediate states of the density, velocity components, and specific total energy. For the backward Euler time stepping, we show the invariant domain preserving (IDP) property of the fully discrete MCL scheme by constructing a fixed-point iteration that meets the requirements of a Krasnoselskii-type theorem. Our iterative solver for the nonlinear discrete problem employs a more efficient fixed-point iteration. The matrix of the associated linear system is a robust low-order Jacobian approximation that exploits the homogeneity property of the flux function. The limited antidiffusive terms are treated explicitly. We use positivity preservation as a stopping criterion for nonlinear iterations. The first iteration yields the solution of a linearized semi-implicit problem. This solution possesses the discrete conservation property but is generally not IDP. Further iterations are performed if any non-IDP states are detected. The existence of an IDP limit is guaranteed by our analysis. To facilitate convergence to steady-state solutions, we perform adaptive explicit underrelaxation at the end of each time step. The calculation of appropriate relaxation factors is based on an approximate minimization of nodal entropy residuals. The performance of proposed algorithms and alternative solution strategies is illustrated by the convergence history for standard two-dimensional test problems.

Monolithic convex limiting and implicit pseudo-time stepping for calculating steady-state solutions of the Euler equations

Abstract

In this work, we use the monolithic convex limiting (MCL) methodology to enforce relevant inequality constraints in implicit finite element discretizations of the compressible Euler equations. In this context, preservation of invariant domains follows from positivity preservation for intermediate states of the density and internal energy. To avoid spurious oscillations, we additionally impose local maximum principles on intermediate states of the density, velocity components, and specific total energy. For the backward Euler time stepping, we show the invariant domain preserving (IDP) property of the fully discrete MCL scheme by constructing a fixed-point iteration that meets the requirements of a Krasnoselskii-type theorem. Our iterative solver for the nonlinear discrete problem employs a more efficient fixed-point iteration. The matrix of the associated linear system is a robust low-order Jacobian approximation that exploits the homogeneity property of the flux function. The limited antidiffusive terms are treated explicitly. We use positivity preservation as a stopping criterion for nonlinear iterations. The first iteration yields the solution of a linearized semi-implicit problem. This solution possesses the discrete conservation property but is generally not IDP. Further iterations are performed if any non-IDP states are detected. The existence of an IDP limit is guaranteed by our analysis. To facilitate convergence to steady-state solutions, we perform adaptive explicit underrelaxation at the end of each time step. The calculation of appropriate relaxation factors is based on an approximate minimization of nodal entropy residuals. The performance of proposed algorithms and alternative solution strategies is illustrated by the convergence history for standard two-dimensional test problems.
Paper Structure (13 sections, 2 theorems, 114 equations, 13 figures)

This paper contains 13 sections, 2 theorems, 114 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Low-order space discretization
  3. Backward Euler time discretization
  4. Monolithic convex limiting
  5. Solution of nonlinear systems
  6. Solver for individual time steps
  7. Steady-state solver
  8. Numerical examples
  9. GAMM channel
  10. Converging-diverging nozzle
  11. NACA 0012 airfoil
  12. Mach 20 bow shock
  13. Conclusions

Key Result

Theorem 1

Let $M$ be a closed, convex, and nonempty subset of a Banach space $(S,\|\cdot\|)$. Suppose that $A:M\to S$ and $B:S\to S$ are such that Then there exists $y\in M$ such that $y=Ay+By$.

Figures (13)

  • Figure 1: GAMM channel ($M_\infty= 0.67$), steady-state MCL results obtained with $N_h = 112,963$ unknowns per component on a mesh consisting of $E_h = 224,576$ triangles.
  • Figure 2: GAMM channel ($M_\infty = 0.67$), steady-state convergence history for a long-time MCL simulation on an unstructured triangular mesh with $N_h = 24,037$ nodes and $E_h = 47,256$ cells.
  • Figure 3: Subsonic nozzle ($M_\infty= 0.2$), steady-state MCL results obtained with $N_h = 95,329$ unknowns per component on a mesh consisting of $E_h = 189,024$ triangles.
  • Figure 5: Transonic nozzle ($M_\infty= 0.3$), steady-state MCL results obtained with $N_h = 95,329$ unknowns per component on a mesh consisting of $E_h = 189,024$ triangles.
  • Figure 6: Transonic nozzle ($M_\infty = 0.3$), steady-state convergence history for a long-time MCL simulation on an unstructured triangular mesh with $N_h = 24,037$ nodes and $E_h = 47,256$ cells.
  • ...and 8 more figures

Theorems & Definitions (10)

  • remark 1
  • Theorem 1: A fixed-point theorem of Krasnoselskii
  • proof
  • Theorem 2
  • proof
  • remark 2
  • remark 3
  • remark 4
  • remark 5
  • remark 6