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Active flux methods for hyperbolic conservation laws -- flux vector splitting and bound-preservation: Two-dimensional case

Junming Duan, Wasilij Barsukow, Christian Klingenberg

Abstract

This paper studies the active flux (AF) methods for two-dimensional hyperbolic conservation laws, focusing on the flux vector splitting (FVS) for the point value update and bound-preserving (BP) limitings, which is an extension of our previous work [J.M. Duan, W. Barsukow, C. Klingenberg, arXiv:2405.02447]. The FVS-based point value update is shown to address the mesh alignment issue that appeared in a quasi-2D Riemann problem along one axis direction on Cartesian meshes. Consequently, the AF methods based on the FVS outperform those using Jacobian splitting, which are prone to transonic and mesh alignment issues. A shock sensor-based limiting is proposed to enhance the convex limiting for the cell average, which can reduce oscillations well. Some benchmark problems are tested to verify the accuracy, BP property, and shock-capturing ability of our BP AF method. Moreover, for the double Mach reflection and forward-facing step problems, the present AF method can capture comparable or better small-scale features compared to the third-order discontinuous Galerkin method with the TVB limiter on the same mesh resolution, while using fewer degrees of freedom, demonstrating the efficiency and potential of our BP AF method for high Mach number flows.

Active flux methods for hyperbolic conservation laws -- flux vector splitting and bound-preservation: Two-dimensional case

Abstract

This paper studies the active flux (AF) methods for two-dimensional hyperbolic conservation laws, focusing on the flux vector splitting (FVS) for the point value update and bound-preserving (BP) limitings, which is an extension of our previous work [J.M. Duan, W. Barsukow, C. Klingenberg, arXiv:2405.02447]. The FVS-based point value update is shown to address the mesh alignment issue that appeared in a quasi-2D Riemann problem along one axis direction on Cartesian meshes. Consequently, the AF methods based on the FVS outperform those using Jacobian splitting, which are prone to transonic and mesh alignment issues. A shock sensor-based limiting is proposed to enhance the convex limiting for the cell average, which can reduce oscillations well. Some benchmark problems are tested to verify the accuracy, BP property, and shock-capturing ability of our BP AF method. Moreover, for the double Mach reflection and forward-facing step problems, the present AF method can capture comparable or better small-scale features compared to the third-order discontinuous Galerkin method with the TVB limiter on the same mesh resolution, while using fewer degrees of freedom, demonstrating the efficiency and potential of our BP AF method for high Mach number flows.
Paper Structure (18 sections, 6 theorems, 68 equations, 18 figures, 1 table)

This paper contains 18 sections, 6 theorems, 68 equations, 18 figures, 1 table.

Table of Contents

  1. Introduction
  2. 2D active flux methods on Cartesian meshes
  3. Point value update using flux vector splitting
  4. Local Lax-Friedrichs flux vector splitting
  5. Upwind flux vector splitting
  6. Van Leer-Hänel flux vector splitting for the Euler equations
  7. Time discretization
  8. Convex limiting for cell average
  9. Application to scalar conservation laws
  10. Application to the compressible Euler equations
  11. Positivity of density
  12. Positivity of pressure
  13. Shock sensor-based limiting
  14. Scaling limiter for point value
  15. Application to scalar conservation laws
  16. ...and 3 more sections

Key Result

Lemma 3.1

For the scalar conservation laws eq:2d_scalar, the intermediate state $\widetilde{u}=\frac{1}{2}(u_L + u_R) + \frac{1}{2\alpha}(f_\ell(u_L) - f_\ell(u_R))$ stays in $\mathcal{G}$eq:2d_scalar_g if $\alpha\geqslant\max\{\varrho_\ell(u_L), \varrho_\ell(u_R)\}$.

Figures (18)

  • Figure 1: The DoFs for the third-order AF method: cell average (circle), face-centered value (squares), nodal value (dots). Note that the cell-centered point value $u_{i,j}$ (cross) is used in constructing the schemes, but does not belong to the DoFs.
  • Figure 2: \ref{['ex:2d_advection_discontinuity']}, 2D advection equation. The numerical solution is obtained based on the JS on the uniform $100\times100$ mesh. From left to right: without any limiting, with BP limitings imposing the global MP, cut-line along $y=x$.
  • Figure 3: \ref{['ex:2d_burgers']}, 2D Burgers' equation. The numerical solution is obtained using the LLF FVS on the uniform $100\times100$ mesh. From left to right: without limiting, with BP limiting, cut-line along $y=x$.
  • Figure 4: \ref{['ex:2d_burgers']}, 2D Burgers' equation. The blending coefficients used in the limitings. From left to right: $\theta_{i+\frac{1}{2},j}$ and $\theta_{i,j+\frac{1}{2}}$ for the cell average, $\theta_{\sigma}$ for the point value.
  • Figure 5: \ref{['ex:2d_vortex']}, 2D isentropic vortex problem. The errors of the conservative variables in the $\ell^1$ norm.
  • ...and 13 more figures

Theorems & Definitions (23)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.1
  • Lemma 3.3
  • proof
  • Proposition 3.1
  • proof
  • Remark 3.2
  • ...and 13 more