The Active Flux scheme is a Finite Volume scheme with additional degrees of freedom. It makes use of a continuous reconstruction and does not require a Riemann solver. An evolution operator is used for the additional degrees of freedom on the cell boundaries. This paper presents progress towards the computation of one-dimensional, viscous, compressible flows using Active Flux scheme. An evolution operator for both linear and nonlinear hyperbolic conservation systems is presented and then a novel extension is made to include source terms. Applications are made on the Euler equations and a hyperbolic formulation of the diffusion equation. Lastly, for the compressible Navier-Stokes equations, a hyperbolic formulation is presented together with a novel operator splitting approach. These allow for the Active Flux evolution operators to be applied to the numerical computation of viscous, compressible flows.

翻译：活性通量格式是一种具有额外自由度的有限体积格式。该格式利用连续重构，无需黎曼求解器。对于单元边界上的额外自由度，采用演化算子进行计算。本文介绍了利用活性通量格式计算一维黏性可压缩流动的进展。首先给出了线性和非线性双曲守恒系统的演化算子，然后提出了一种新颖的扩展方法以包含源项。接着将该方法应用于欧拉方程和扩散方程的双曲型公式中。最后，针对可压缩纳维-斯托克斯方程，提出了一种双曲型公式及其新型算子分裂方法。这些工作使得活性通量演化算子能够应用于黏性可压缩流动的数值计算。