Active Flux is an extension of the Finite Volume method and additionally incorporates point values located at cell boundaries. This gives rise to a globally continuous approximation of the solution. The method is third-order accurate. We demonstrate that a new semi-discrete Active Flux method (first described in Abgrall&Barsukow, 2023 for one space dimension) can easily be used to solve nonlinear hyperbolic systems in multiple dimensions, such as the compressible Euler equations of inviscid hydrodynamics. Originally, the Active Flux method emerged as a fully discrete method, and required an exact or approximate evolution operator for the point value update. For nonlinear problems such an operator is often difficult to obtain, in particular for multiple spatial dimensions. With the new approach it becomes possible to leave behind these difficulties. We introduce a multi-dimensional limiting strategy and demonstrate the performance of the new method on both Riemann problems and subsonic flows.

翻译：Active Flux是有限体积法的推广，其额外引入了位于单元边界的点值。这构成了解的全局连续近似。该方法具有三阶精度。我们证明了一种新的半离散Active Flux方法（最初由Abgrall与Barsukow于2023年在一维空间中描述）可简便地用于求解多维非线性双曲型方程组，例如无粘流体动力学中可压缩欧拉方程。最初提出的Active Flux方法为全离散方法，需要精确或近似的演化算子来更新点值。对于非线性问题，此类算子往往难以获取，尤其是在多维空间中。采用新方法后，这些困难得以克服。我们引入了一种多维限制策略，并通过黎曼问题及亚音速流实例展示了新方法的性能。