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The solution of time-dependent hyperbolic conservation laws on cut cellmeshes causes the small cell problem: standard schemes are not stable on thearbitrarily small cut cells if an explicit time stepping scheme is used and thetime 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 generaland MUSCL-Trap in particular have been introduced to solve this problem byusing implicit time stepping on the cut cells. Theoretical and numericalresults have indicated that this might lead to a loss in accuracy whenswitching between the explicit and implicit time stepping. In this contributionwe examine this in more detail and will prove in one dimension that thespecific combination MUSCL-Trap of an explicit second-order and an implicitsecond-order scheme results in a fully second-order mixed scheme. As thisresult is unlikely to hold in two dimensions, we also introduce two newversions of mixed explicit implicit schemes based on exchanging the explicitscheme. We present numerical tests in two dimensions where we compare the newversions with the original MUSCL-Trap scheme.

Accuracy analysis for explicit-implicit finite volume schemes on cut cell meshes