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A compact high-order and positivity-preserving active flux method for compressible Navier--Stokes equations on Cartesian meshes

Junming Duan, Wasilij Barsukow, Christian Klingenberg

arXiv:2609.29081v1math.NAcs.NA

TL;DR

Compact fourth-order active flux methods for compressible Navier–Stokes equations face challenges in accurately discretizing diffusion and preserving physical states. This paper develops a positivity-preserving method with compact fourth-order viscous operators and a joint flux limiter. In the viscous shock tube comparison, AF is approximately ten times faster than DG at comparable accuracy and with the same number of degrees of freedom.

  • Problem

    For compressible Navier–Stokes equations, standard third-order active flux operators can reduce diffusion accuracy to second order in 2D, while strong shocks and near-vacuum states make positivity-preserving limiting delicate.

    Source: §1 · pp. 2–3

  • Method

    The method retains standard third-order AF degrees of freedom, uses compact fourth-order operators for viscous flux divergence, and jointly limits inviscid and viscous fluxes with a point-value scaling limiter.

    Source: §1 · p. 3

  • Results

    For the viscous shock tube, AF is approximately ten times faster than DG at comparable accuracy and with the same number of degrees of freedom.

    Source: §1 · p. 3

  • Takeaways & Limitations

    The reported shock-tube comparison shows that this compact AF method can resolve shock–boundary-layer interaction with comparable accuracy to DG while using less wall-clock time and fewer degrees of freedom.

    Source: §5 · pp. 22–23

    The stated formulation uses an ideal-gas equation of state and assumes constant dynamic viscosity.

    Source: §1 · p. 2

Abstract

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This paper develops a compact fourth-order positivity-preserving active flux (AF) method for the one- and two-dimensional compressible Navier--Stokes equations on Cartesian meshes. The method retains the cell averages and shared point values of the standard third-order AF method as its degrees of freedom. To avoid the order reduction that can arise when diffusion is discretized using operators from the standard third-order AF method, while maintaining compactness, the divergence of the viscous flux is discretized directly using compact fourth-order operators. For the inviscid part, incorporating a downwind point value into the biased stencil yields fourth-order accuracy. A monolithic flux limiting blends high-order total numerical fluxes with low-order positivity-preserving counterparts, treating the inviscid and viscous fluxes jointly while maintaining local conservation. Together with a scaling limiter for point values, this procedure preserves density and pressure positivity for both cell averages and point values. Numerical experiments demonstrate fourth-order convergence, positivity preservation, and accurate resolution of shocks and viscous flow structures. For the two-dimensional viscous shock tube, a comparison with a discontinuous Galerkin method shows improved computational efficiency in resolving complex interaction of shock waves and boundary layers.

1 Introduction

The introduction motivates compact, high-order methods for compressible Navier–Stokes equations and identifies diffusion accuracy and positivity preservation as challenges for active flux methods. The paper addresses these challenges with a compact fourth-order positivity-preserving method, reporting an efficiency advantage over DG for the viscous shock tube.

1 Introduction

  • High-order methods can resolve smooth and multiscale flow structures on coarse meshes, while compact, local data dependencies support communication-efficient parallel implementation.The introduction notes that tensor-product DG degrees of freedom grow quadratically and cubically with polynomial degree in 2D and 3D, respectively.Source: §1 · p. 2
  • Prior AF diffusion approaches include auxiliary-variable formulations and direct gradient approximations, while the latter is second-order accurate; standard third-order operators can also reduce to second order in 2D.The paper motivates direct compact fourth-order discretization of viscous flux divergence to address this accuracy issue.Source: §1 · p. 3
  • Strong shocks and near-vacuum states make positivity preservation important, and compressible Navier–Stokes viscous fluxes complicate limiting through gradient-dependent, coupled stress components.The method targets density and pressure positivity for both cell averages and point values.Source: §1 · p. 3
  • For the viscous shock tube, AF is approximately ten times faster than DG at comparable accuracy and with the same number of DoFs.The comparison concerns the test with shock–boundary-layer interaction.Source: §1 · p. 3
  • The method retains the standard third-order AF degrees of freedom while using compact fourth-order operators for viscous flux divergence and a downwind point value for fourth-order inviscid accuracy.A monolithic limiter jointly limits inviscid and viscous fluxes against a low-order positivity-preserving flux; a scaling limiter treats point values.Source: §1 · p. 3

2 The compact AF formulation

The compact AF formulation evolves cell averages and shared point values, combining biased inviscid operators with direct compact discretizations of viscous flux divergence. Its 2D extension uses Cartesian-mesh degrees of freedom and directional operators, with SSP-RK54 time integration.

2.1 The 1D formulation

  • In 1D, the cell-average update is conservative and the degrees of freedom are cell averages and shared interface point values.The equation discretizes the viscous flux divergence directly as ∂x(K∂xZ).Source: §2.1 · pp. 4–5
  • In 1D, local Lax–Friedrichs flux-vector splitting supplies the inviscid point-value update, whose biased operators use cubic reconstructions for fourth-order accuracy.The symmetric average of biased stencils gives central differences for viscous gradients; the diffusion part has no upwinding mechanism.Source: §2.1 · pp. 4–5
  • The 1D viscous discretization uses a compact fourth-order second-derivative operator, with coefficients selected to satisfy order conditions, sparsity, and symmetry.The right-hand side discretizes ∂x(K∂xZ) directly.Source: §2.1 · p. 5

2.2 The 2D formulation

  • In 2D, the method uses a uniform Cartesian mesh with cell averages, edge-centered values, and corner point values as degrees of freedom.The state includes density, momentum components, and energy; the auxiliary variables include velocity components and temperature.Source: §2.2 · p. 6
  • In 2D, edge flux averages are approximated by Simpson’s rule, and mixed derivatives in the point-value update are discretized by applying directional operators sequentially.The finite-difference operators are applied along the coordinate directions.Source: §2.2 · p. 6
  • The 2D formulation obtains cell-centered point values through a separate reconstruction and uses a central finite-difference operator.Source: §2.2 · p. 7

2.3 Time discretization

  • The semi-discretizations use the explicit five-stage, fourth-order SSP-RK54 method for time integration.Source: §2.3 · p. 7
  • For 1D linear advection, the stable CFL limit exceeds that of the standard third-order AF method, an advantage attributed to the additional downwind point in the biased stencil.Source: §2.3 · p. 7

3 Positivity-preserving limiting and oscillation control

The limiting framework uses admissible intermediate states to establish positivity for inviscid and viscous fluxes, then combines them in the final procedure. It applies smoothness-based and positivity-based limiting to cell averages and point values.

3 Positivity-preserving limiting and oscillation control

  • The framework proves positivity separately for inviscid and viscous split states, then combines them in the final limiting procedure.The proof uses intermediate states in the convex admissible set.Source: §3 · p. 8, §3.1 · p. 9

3.1 Preliminaries

  • Convex decompositions of intermediate states provide the basis for positivity-preserving low-order fluxes and the combined limiting framework.The cell-average update uses a low-order LLF scheme whose flux blending retains local conservation.Source: §3.1 · pp. 9–10

3.3 Positivity-preserving limiting

  • Cell-average limiting blends high-order fluxes with low-order LLF counterparts and constrains density and pressure to preserve positivity through SSP-RK stages.The low-order blending preserves local conservation; applying the limiter at every forward-Euler stage preserves positivity through convex stage combinations.Source: §3.1 · p. 10, §3.3 · p. 12

3.2 Smoothness indicator based limiting

  • The smoothness-based limiter uses a sensor combining Jameson’s shock sensor and a modified Ducros sensor, with central differences for velocity divergence and curl.The strength parameter Cs is fixed at 1.Source: §3.2 · pp. 10–11

3.4 Limiting for the point value

  • A scaling limiter blends high-order point values with a low-order positivity-preserving LLF solution to preserve density and pressure positivity under the stated time-step conditions.The stated result applies to both cell averages and point values.Source: §3.4 · p. 13

4 Numerical experiments

The experiments test the proposed AF method on smooth, shocked, and viscous flows in one and two dimensions. They report fourth-order convergence, resolved flow structures, and an efficiency advantage over DG in the viscous shock-tube comparison.

Example 4.1: 1D viscous shock wave

  • The AF method captures the 1D viscous shock accurately and achieves fourth-order convergence under mesh refinement for both tested cases.The two cases use M0 = 3 and M0 = 20.Source: §4 · p. 14, Figure 1 · p. 14, Figure 2 · p. 15

Example 4.2: 2D manufactured Taylor–Green vortex

Example 4.3: First problem of Stokes

  • For the Stokes diffusion test, the AF solutions agree well with reference solutions without oscillations across three tested viscosity values.The tested values are µ = 10^-2, 10^-3, and 10^-4.Source: §4 · p. 15, Figure 5 · p. 17

Example 4.4: One-dimensional LeBlanc problem

  • In the LeBlanc test, viscosity smooths discontinuities; the inviscid solution has a small density undershoot near the rarefaction tail, while both solutions otherwise agree with references.The inviscid result is compared with the exact Euler solution and the viscous result with a finer AF reference.Source: §4 · pp. 16–17, Figure 6 · p. 17

Example 4.5: Shock diffraction

  • For shock diffraction, the AF results agree well with prior results, and viscosity produces a smoother solution near the corner.The comparison contrasts the inviscid case with the case at Re = 200.Source: §4 · p. 18, Figure 7 · p. 18

Example 4.6: Two-dimensional viscous shock tube

  • At equal DoFs, AF captures shock waves and small structures better than DG in the viscous shock tube, and agrees better with the reference solution.The comparison reports sharper lambda-shock features and structures near the wedge-shaped region.Source: §4 · pp. 18–19, Figure 8 · p. 19
  • The AF method is more efficient than DG at the same DoF count and gives better wall-density resolution at the same wall-clock time.The comparison uses the wall-density ℓ1 error versus wall-clock time; Figure 10 presents the comparison.Source: §4 · p. 20, Figure 10 · p. 20

Examples 4.7–4.8: Low-Mach viscous flows

  • The cavity test captures the main recirculation and secondary corner vortices, with velocity profiles matching reference solutions; the double-shear-layer vorticity also agrees well with prior results.The cavity comparison covers Re = 100 and 1000; the double-shear-layer test is at Re = 10000.Source: §4 · p. 21, Figure 11 · p. 21, Figure 12 · p. 22

5 Conclusions

The paper concludes that its compact fourth-order positivity-preserving active-flux method handles one- and two-dimensional compressible Navier–Stokes equations on Cartesian meshes. In the two-dimensional viscous shock tube, it achieves comparable accuracy to DG with fewer degrees of freedom and less wall-clock time.

5 Conclusions

  • The method achieves fourth-order accuracy for the inviscid discretization and uses direct compact fourth-order operators for the viscous flux divergence.It retains the degrees of freedom of the standard third-order AF method.Source: §5 · p. 22
  • Monolithic flux limiting treats inviscid and viscous fluxes jointly while maintaining local conservation.Together with a scaling limiter for point values, it preserves density and pressure positivity and suppresses nonphysical oscillations near strong discontinuities.Source: §5 · p. 22
  • Experiments demonstrate fourth-order convergence for smooth viscous flows and positivity preservation in problems involving strong discontinuities.Source: §5 · p. 22
  • The added downwind point in the biased stencil makes the CFL condition less restrictive than in the standard third-order AF method.Source: §5 · p. 22
  • In the two-dimensional viscous shock tube, AF attains comparable accuracy to DG with fewer degrees of freedom and less wall-clock time.The comparison concerns shock–boundary-layer interaction.Source: §5 · pp. 22–23
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