We extend the fourth order, two stage Multi-Derivative Runge Kutta (MDRK) scheme of Li and Du to the Flux Reconstruction (FR) framework by writing both of the stages in terms of a time averaged flux and then use the approximate Lax-Wendroff procedure. Numerical flux is computed in each stage using D2 dissipation and EA flux, enhancing Fourier CFL stability and accuracy respectively. A subcell based blending limiter is developed for the MDRK scheme, which ensures that the limited scheme is provably admissibility preserving. Along with being admissibility preserving, the blending scheme is constructed to minimize dissipation errors by using Gauss-Legendre solution points and performing MUSCL-Hancock reconstruction on subcells. The accuracy enhancement of the blending scheme is numerically verified on compressible Euler's equations, with test cases involving shocks and small-scale structures.

中文翻译：

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双曲守恒定律的多导数龙格-库塔通量重建

我们将 Li 和 Du 的四阶两阶段多导数龙格库塔 (MDRK) 方案扩展到通量重建 (FR) 框架，将两个阶段都写成时间平均通量，然后使用近似的 Lax-Wendroff程序。使用 D2 耗散和 EA 通量计算每个阶段的数值通量，分别增强傅里叶 CFL 的稳定性和准确性。为 MDRK 方案开发了基于子单元的混合限制器，这确保了有限方案可证明保持可接受性。除了保留可允许性之外，还构建了混合方案，通过使用 Gauss-Legendre 解点并对子单元执行 MUSCL-Hancock 重建来最小化耗散误差。混合方案的精度增强在可压缩欧拉方程上进行了数值验证，测试用例涉及冲击和小规模结构。