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.

翻译：活动通量（AF）方法是一种紧致高阶有限体积方法，可同时演化单元平均值和单元界面处的点值。在线法框架内，现有基于雅可比分裂的点值更新方法虽融入了迎风思想，但由于对迎风方向估计不准确，在非线性问题中存在停滞现象，同时部分由于解耦的点值更新导致网格对齐问题。本文提出在点值更新中采用通量矢量分裂，为这两个问题提供了自然且统一的解决方案。为提升鲁棒性，本文还针对双曲守恒律发展了保界（BP）AF方法。考虑两种情形：标量情形的最大值原理保持，以及可压缩欧拉方程的正密度与正压力保持。通过将单元平均值更新重写为原始高阶通量与鲁棒低阶（局部Lax-Friedrichs或Rusanov）通量的凸组合，并通过选择恰当比例的低阶通量来强制实现期望的界。点值更新采用类似的混合策略。此外，提出基于激波探测器的限制器以增强单元平均值的凸限制，能有效抑制振荡。通过包括前向台阶绕流和高马赫数射流在内的多项挑战性测试，验证了BP AF方法的鲁棒性与有效性。