双曲守恒律的主动通量方法——通量向量分裂与保界性：一维情形
Active flux methods for hyperbolic conservation laws -- flux vector splitting and bound-preservation: One-dimensional case
AI总结：
针对基于雅可比分裂的主动通量方法存在的跨声速问题，本文提出用通量向量分裂更新点值，并开发了一维双曲守恒律的保界主动通量方法，通过高低阶通量凸组合实现界约束，经基准测试验证了其精度、保界性与激波捕捉能力。
摘要：
主动通量（active flux, AF）方法是一种紧致的高阶有限体积方法，可独立演化单元平均值和单元界面处的点值。在线法框架内，点值可基于雅可比分裂（Jacobian splitting, JS）进行更新，融入了迎风思想。然而，这类基于JS的AF方法在处理非线性问题时会因迎风方向估计不准确而遭遇跨声速问题。本文提出将通量向量分裂用于点值更新，为跨声速问题提供了一种自然且统一的解决方案。为提升鲁棒性，本文还针对一维双曲守恒律开发了保界（bound-preserving, BP）AF方法，考虑了两种情形：标量情形下的最大值原理保持，以及可压缩Euler方程的正密度与正压力保持。本文将高阶AF方法中的单元平均值更新改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，通过选择适量的低阶通量来施加所需的界约束；点值更新也采用了类似的混合策略。本文开展了若干具有挑战性的基准测试，以验证该方法的精度、保界性质和激波捕捉能力。
英文摘要：
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.
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