The solution of time-dependent hyperbolic conservation laws on cut cell meshes causes the small cell problem: standard schemes are not stable on the arbitrarily small cut cells if an explicit time stepping scheme is used and the time step size is chosen based on the size of the background cells. In [J. Sci. Comput. 71, 919-943 (2017)], the mixed explicit implicit approach in general and MUSCL-Trap in particular have been introduced to solve this problem by using implicit time stepping on the cut cells. Theoretical and numerical results have indicated that this might lead to a loss in accuracy when switching between the explicit and implicit time stepping. In this contribution we examine this in more detail and will prove in one dimension that the specific combination MUSCL-Trap of an explicit second-order and an implicit second-order scheme results in a fully second-order mixed scheme. As this result is unlikely to hold in two dimensions, we also introduce two new versions of mixed explicit implicit schemes based on exchanging the explicit scheme. We present numerical tests in two dimensions where we compare the new versions with the original MUSCL-Trap scheme.

翻译：对于定义在切割网格上的时间依赖型双曲守恒律，存在小网格问题：当采用显式时间推进格式且时间步长基于背景网格尺寸选取时，标准格式在任意小的切割网格上不稳定。文献[J. Sci. Comput. 71, 919-943 (2017)]提出了混合显式-隐式方法（特别是MUSCL-Trap格式），通过对切割网格采用隐式时间推进解决该问题。理论与数值结果表明，显式与隐式时间推进之间的切换可能导致精度损失。本文对此进行更深入研究，并在一维情形下证明：显式二阶格式与隐式二阶格式的特定组合MUSCL-Trap能形成完全二阶精度的混合格式。鉴于该结论在二维情形下难以成立，我们基于显式格式的替换提出两种新型混合显式-隐式格式。通过二维数值测试，将新格式与原始MUSCL-Trap格式进行对比分析。