Lax-Wendroff Flux Reconstruction (LWFR) is a single-stage, high order, quadrature free method for solving hyperbolic conservation laws. We develop a subcell based limiter by blending LWFR with a lower order scheme, either first order finite volume or MUSCL-Hancock scheme. While the blending with a lower order scheme helps to control oscillations, it may not guarantee admissibility of discrete solution, e.g., positivity property of quantities like density and pressure. By exploiting the subcell structure and admissibility of lower order schemes, we devise a strategy to ensure that the blended scheme is admissibility preserving for the mean values and then use a scaling limiter to obtain admissibility of the polynomial solution. For MUSCL-Hancock scheme on non-cell-centered subcells, we develop a slope limiter, time step restrictions and suitable blending of higher order fluxes, that ensures admissibility of lower order updates and hence that of the cell averages. By using the MUSCL-Hancock scheme on subcells and Gauss-Legendre points in flux reconstruction, we improve small-scale resolution compared to the subcell-based RKDG blending scheme with first order finite volume method and Gauss-Legendre-Lobatto points. We demonstrate the performance of our scheme on compressible Euler's equations, showcasing its ability to handle shocks and preserve small-scale structures.

翻译：Lax-Wendroff通量重构（LWFR）是一种单步、高阶、免求积的求解双曲守恒律的方法。我们通过将LWFR与低阶格式（一阶有限体积法或MUSCL-Hancock格式）混合，开发了一种基于子网格的限制器。尽管与低阶格式混合有助于控制振荡，但无法保证离散解的容许性，例如密度和压强等量的正性保持。通过利用子网格结构及低阶格式的容许性，我们设计了一种策略，确保混合格式对均值具有保容许性，并进一步采用缩放限制器获得多项式解的容许性。针对非单元中心的MUSCL-Hancock格式子网格，我们开发了斜率限制器、时间步长限制及高阶通量的适当混合方法，从而确保低阶更新的容许性，进而保证单元平均值的容许性。通过在子网格上使用MUSCL-Hancock格式并结合通量重构中的Gauss-Legendre点，与基于一阶有限体积法和Gauss-Legendre-Lobatto点的RKDG子网格混合格式相比，我们提升了小尺度分辨率。我们在可压缩欧拉方程上展示了该格式的性能，体现了其处理激波并保持小尺度结构的能力。