  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.
