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Active flux methods for hyperbolic conservation laws -- flux vector splitting and bound-preservation

Junming Duan, Wasilij Barsukow, Christian Klingenberg

Abstract

The active flux (AF) method is a compact high-order finite volume method that simultaneously evolves cell averages and point values at cell interfaces. Within the method of lines framework, the existing Jacobian splitting-based point value update incorporates the upwind idea but suffers from a stagnation issue for nonlinear problems due to inaccurate estimation of the upwind direction, and also from a mesh alignment issue partially resulting from decoupled point value updates. This paper proposes to use flux vector splitting for the point value update, offering a natural and uniform remedy to those two issues. To improve robustness, this paper also develops bound-preserving (BP) AF methods for hyperbolic conservation laws. Two cases are considered: preservation of the maximum principle for the scalar case, and preservation of positive density and pressure for the compressible Euler equations. The update of the cell average is rewritten as a convex combination of the original high-order fluxes and robust low-order (local Lax-Friedrichs or Rusanov) fluxes, and the desired bounds are enforced by choosing the right amount of low-order fluxes. A similar blending strategy is used for the point value update. In addition, a shock sensor-based limiting is proposed to enhance the convex limiting for the cell average, which can suppress oscillations well. Several challenging tests are conducted to verify the robustness and effectiveness of the BP AF methods, including flow past a forward-facing step and high Mach number jets.

Active flux methods for hyperbolic conservation laws -- flux vector splitting and bound-preservation

Abstract

The active flux (AF) method is a compact high-order finite volume method that simultaneously evolves cell averages and point values at cell interfaces. Within the method of lines framework, the existing Jacobian splitting-based point value update incorporates the upwind idea but suffers from a stagnation issue for nonlinear problems due to inaccurate estimation of the upwind direction, and also from a mesh alignment issue partially resulting from decoupled point value updates. This paper proposes to use flux vector splitting for the point value update, offering a natural and uniform remedy to those two issues. To improve robustness, this paper also develops bound-preserving (BP) AF methods for hyperbolic conservation laws. Two cases are considered: preservation of the maximum principle for the scalar case, and preservation of positive density and pressure for the compressible Euler equations. The update of the cell average is rewritten as a convex combination of the original high-order fluxes and robust low-order (local Lax-Friedrichs or Rusanov) fluxes, and the desired bounds are enforced by choosing the right amount of low-order fluxes. A similar blending strategy is used for the point value update. In addition, a shock sensor-based limiting is proposed to enhance the convex limiting for the cell average, which can suppress oscillations well. Several challenging tests are conducted to verify the robustness and effectiveness of the BP AF methods, including flow past a forward-facing step and high Mach number jets.

Paper Structure

This paper contains 36 sections, 9 theorems, 123 equations, 28 figures, 1 table.

Table of Contents

  1. Introduction
  2. 1D active flux methods
  3. Stagnation issue when using Jacobian splitting
  4. Point value update using flux vector splitting
  5. Local Lax-Friedrichs flux vector splitting
  6. Upwind flux vector splitting
  7. Van Leer-Hänel flux vector splitting for the Euler equations
  8. Time discretization
  9. 2D active flux methods on Cartesian meshes
  10. Point value update using flux vector splitting
  11. Mesh alignment issue when using Jacobian splitting
  12. Boundary treatment
  13. 2D bound-preserving active flux methods
  14. Convex limiting for the cell average
  15. Application to scalar conservation laws
  16. ...and 21 more sections

Key Result

Lemma 4.1

If the time step size $\Delta t^n$ satisfies then eq:2d_lo_decomp is a convex combination, and the first-order LLF scheme is BP.

Figures (28)

  • Figure 1: The DoFs for the third-order AF method: cell average (circle), face-centered values (squares), values at corners (dots). Note that the cell-centered point value $u_{i,j}$ (cross) is used in constructing the scheme, but does not belong to the DoFs.
  • Figure 2: The stencils for the first-order LLF schemes.
  • Figure 3: \ref{['ex:1d_burgers']}, self-steepening shock for the Burgers' equation. The numerical solutions computed without limiting (top row) and with the BP limitings imposing the local MP (bottom row). From left to right: JS, LLF, and upwind FVS.
  • Figure 4: \ref{['ex:1d_leblanc']}, LeBlanc Riemann problem. The density computed with the BP limitings and the shock sensor-based limiting ($\kappa=10$) on a uniform mesh of $6000$ cells. From left to right: JS, LLF, SW, and VH FVS.
  • Figure 5: \ref{['ex:1d_blast_wave']}, blast wave interaction. The density computed with the BP limitings and the shock sensor-based limiting ($\kappa=1$). The corresponding enlarged views in $x\in[0.62, 0.82]$ are shown in the bottom row.
  • ...and 23 more figures

Theorems & Definitions (39)

  • Remark 2.1
  • Remark 2.2
  • Definition 4.1
  • Lemma 4.1
  • Proposition 4.1
  • proof
  • Remark 4.1
  • Remark 4.2
  • Lemma 4.2
  • Remark 4.3
  • ...and 29 more