High-order schemes with larger time-step limits can benefit from using adaptive mesh refinement grids. One-parameter optimal flux reconstruction schemes are obtained by minimizing the error associated with wave propagation over the range of resolvable wavenumbers using a novel objective function that accounts for the Courant–Friedrichs–Lewy limit and order of accuracy constraint. The novel schemes are parameterized by the Courant–Friedrichs–Lewy limit, which, if chosen properly, leads to the obtained optimal Courant–Friedrichs–Lewy-like well-known schemes. The schemes are obtained by the differential evolution optimization algorithm, which is combined with the flux reconstruction–discontinuous Galerkin scheme to enlarge the time step limit on adaptive mesh refinement grids. Two numerical experiments were performed to investigate the properties of the schemes, including advection of a Gaussian bump and isentropic Euler vortex. The results show that the new schemes are more suitable for adaptive mesh refinement in terms of accuracy, dissipation, and stability.

具有较大时间步长限制的高阶方案可以从使用自适应网格细化网格中受益。单参数最优通量重建方案是通过使用一种新的目标函数来最小化与可解析波数范围内的波传播相关的误差来获得的，该函数考虑了 Courant-Friedrichs-Lewy 极限和精度约束的顺序。新方案由 Courant-Friedrichs-Lewy 限制参数化，如果选择得当，会导致获得最佳的 Courant-Friedrichs-Lewy 类众所周知的方案。这些方案是通过差分进化优化算法获得的，该算法结合通量重建-不连续伽辽金方案来扩大自适应网格细化网格的时间步长限制。进行了两个数值实验来研究方案的性质，包括高斯颠簸和等熵欧拉涡流的平流。结果表明，新方案在精度、耗散和稳定性方面更适合自适应网格细化。