Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Junming Duan, Wasilij Barsukow, Christian Klingenberg

Abstract

The active flux (AF) method is a compact high-order finite volume method that evolves cell averages and point values at cell interfaces independently. Within the method of lines framework, the point value can be updated based on Jacobian splitting (JS), incorporating the upwind idea. However, such JS-based AF methods encounter transonic issues for nonlinear problems due to inaccurate upwind direction estimation. This paper proposes to use flux vector splitting for the point value update, offering a natural and uniform remedy to the transonic issue. To improve robustness, this paper also develops bound-preserving (BP) AF methods for one-dimensional 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 in high-order AF methods is rewritten as a convex combination of using 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. Several challenging benchmark tests are conducted to verify the accuracy, BP properties, and shock-capturing ability of the methods.

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

Abstract

The active flux (AF) method is a compact high-order finite volume method that evolves cell averages and point values at cell interfaces independently. Within the method of lines framework, the point value can be updated based on Jacobian splitting (JS), incorporating the upwind idea. However, such JS-based AF methods encounter transonic issues for nonlinear problems due to inaccurate upwind direction estimation. This paper proposes to use flux vector splitting for the point value update, offering a natural and uniform remedy to the transonic issue. To improve robustness, this paper also develops bound-preserving (BP) AF methods for one-dimensional 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 in high-order AF methods is rewritten as a convex combination of using 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. Several challenging benchmark tests are conducted to verify the accuracy, BP properties, and shock-capturing ability of the methods.
Paper Structure (22 sections, 5 theorems, 64 equations, 18 figures, 1 table)

This paper contains 22 sections, 5 theorems, 64 equations, 18 figures, 1 table.

Table of Contents

  1. Introduction
  2. 1D active flux methods for hyperbolic conservation laws
  3. Point value update 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. Convex limiting for the cell average
  10. Application to scalar conservation laws
  11. Application to the compressible Euler equations
  12. Positivity of density
  13. Positivity of pressure
  14. Scaling limiter for point value
  15. Application to scalar conservation laws
  16. ...and 7 more sections

Key Result

Lemma 3.1

If the time step size $\Delta t^n$ satisfies then eq:1d_lo_decomp is a convex combination, and the first-order LLF scheme is BP.

Figures (18)

  • Figure 1: \ref{['ex:1d_advection_discontinuity']}, advection. The results are obtained without any limiting (upper left), with power law reconstruction (upper right), with BP limitings imposing global MP for the cell average and point value (lower left), with BP limitings imposing local MP for the cell average and point value (lower right).
  • Figure 2: \ref{['ex:1d_burgers']}, self-steepening shock for the Burgers' equation. The numerical solutions are based on the JS. From left to right: without limiting, with the power law reconstruction, with the BP limitings imposing local MP for the cell average and point value update, respectively.
  • Figure 3: \ref{['ex:1d_burgers']}, self-steepening shock for the Burgers' equation. From left to right: the LLF FVS without limiting, the LLF FVS with limitings, the SW FVS without limiting, the SW FVS with limitings. The limitings consider the local MP for the cell average and point value updates, respectively.
  • Figure 4: \ref{['ex:1d_accuracy']}, the accuracy test for the 1D Euler equations.
  • Figure 5: \ref{['ex:1d_accuracy']}, the density (left) and velocity (right) are obtained with the SW FVS and $80$ cells for the 1D Euler equations.
  • ...and 13 more figures

Theorems & Definitions (24)

  • Remark 2.1
  • Definition 3.1
  • Remark 3.1
  • Lemma 3.1: Guermond and Popov Guermond_2016_Invariant_SJoNA
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Remark 3.3
  • Lemma 4.1
  • Remark 4.1
  • ...and 14 more