An Explicit Implicit Scheme for Cut Cells in Embedded Boundary Meshes

We present a new mixed explicit implicit time stepping scheme for solving the linear advection equation on a Cartesian cut cell mesh. We use a standard second-order explicit scheme on Cartesian cells away from the embedded boundary. On cut cells, we use an implicit scheme for stability. This approach overcomes the small cell problem—that standard schemes are not stable on the arbitrarily small cut cells—while keeping the cost fairly low. We examine several approaches for coupling the schemes in one dimension. For one of them, which we refer to as flux bounding, we can show a TVD result for using a first-order implicit scheme. We also describe a mixed scheme using a second-order implicit scheme and combine both mixed schemes by an FCT approach to retain monotonicity. In the second part of this paper, extensions of the second-order mixed scheme to two and three dimensions are discussed and the corresponding numerical results are presented. These indicate that this mixed scheme is second-order accurate in \(L^1\) and between first- and second-order accurate along the embedded boundary in two and three dimensions.