The Active Flux scheme is a finite volume scheme with additional point values distributed along the cell boundary. It is third order accurate and does not require a Riemann solver. Instead, given a reconstruction, the initial value problem at the location of the point value is solved. The intercell flux is then obtained from the evolved values along the cell boundary by quadrature. Whereas for linear problems an exact evolution operator is available, for nonlinear problems one needs to resort to approximate evolution operators. This paper presents such approximate operators for nonlinear hyperbolic systems in one dimension and nonlinear scalar equations in multiple spatial dimensions. They are obtained by estimating the wave speeds to sufficient order of accuracy. Additionally, an entropy fix is introduced and a new limiting strategy is proposed. The abilities of the scheme are assessed on a variety of smooth and discontinuous setups.
翻译:主动通量计划是一个有限量计划,在单元格边界沿线分配额外的点值,这是第三顺序准确,不需要里曼求解器。相反,在进行重建后,点值位置的初始值问题得到解决。随后,细胞间通量通过二次曲线从细胞边界沿细胞边界的演变值中获取。对于线性问题,有一个精确的进化操作员,对于非线性问题,需要求助于近似进化操作员。本文介绍了非线性双曲系统在一个维度上的大致操作员和多个空间维度的非线性二次方程的大致操作员。它们是通过估计波速以足够精确的顺序获得的。此外,还引入了一种增缩修正,并提出了新的限制战略。对该计划的能力进行了各种平稳和不连续的设置评估。