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

TL;DR

This paper introduces flux vector splitting into active flux methods for hyperbolic conservation laws, improving robustness and bound-preservation, and demonstrates effectiveness through challenging numerical tests.

Contribution

It proposes a flux vector splitting approach for active flux methods and develops bound-preserving strategies to enhance robustness for nonlinear hyperbolic problems.

Findings

Flux vector splitting remedies stagnation and mesh alignment issues.

Bound-preserving methods maintain physical bounds in simulations.

Numerical tests confirm robustness and accuracy of the proposed methods.

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.