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

TL;DR

This paper introduces flux vector splitting into active flux methods for one-dimensional hyperbolic conservation laws, addressing transonic issues and developing bound-preserving techniques to enhance robustness and accuracy.

Contribution

It proposes a flux vector splitting approach for active flux methods and develops bound-preserving strategies for scalar and Euler equations, improving stability and physical bounds.

Findings

Flux vector splitting remedies transonic issues.

Bound-preserving methods maintain maximum principle and positivity.

Numerical tests confirm accuracy and robustness.

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.