Accuracy considerations of mixed explicit implicit schemes for embedded boundary meshes

We consider a mixed explicit implicit finite volume scheme for solving the advection equation on a Cartesian cut cell mesh: the scheme couples an implicit time stepping scheme on cut cells with an explicit time stepping scheme on cells away from the cut cells by means of flux bounding. We examine the accuracy of the resulting mixed scheme to see if a second‐order explicit and a second‐order implicit scheme result in a second‐order mixed scheme.


The mixed explicit implicit scheme
For flow simulations in complex geometry, the use of cut cell meshes is typically cheaper and faster than employing bodyfitted grids. In this approach the geometry is simply cut out of a Cartesian background grid. This results in so called cut cells around the boundary of the embedded object. These cells have irregular shape and may be very small. In the context of solving time-dependent, hyperbolic partial differential equations, probably the biggest issue caused by this approach is the small cell problem -standard explicit time stepping schemes are not stable on the arbitrarily small cut cells if the time step is chosen according to the background mesh.
The mixed explicit implicit scheme solves this problem by using implicit time stepping on cut cells and explicit time stepping otherwise. In space, a first-or second-order finite volume scheme is used. In order to ensure mass conservation and stability, the time stepping schemes are coupled using so called flux bounding [1]. While the mixed scheme also works well in 2D and 3D, we will focus on 1D for the rest of this contribution for simplicity. For an extended discussion of the scheme and its properties and its accuracy in 2D (and 3D), we refer the reader to [1,2].
We consider the linear advection equation on a bounded interval I with periodic boundary conditions. To examine the behavior of cut cells in 1D, we use the following model problem: we consider a uniform mesh on I with mesh width h, which contains one cell of length αh with α ∈ (0, 1]. This cell is labeled as cell '0'. The idea of flux bounding is illustrated in figure 1. The important aspect is how the explicit and the implicit scheme are coupled: by reusing in Step 2 at edges between cells −2 and −1 and between cells 1 and 2, respectively, the flux from the explicit scheme (that was used in the updates of cells -2 and 2 in Step 1) one ensures mass conservation. Further, for coupling the explicit second-order MUSCL scheme with implicit Euler time stepping with piecewise constant data, one can show a TVD stability result that is independent of the size of the volume fraction α [1].
Step 1: Update of explicit cells.
Step 2: Update of cut cell and transition cells For obtaining a second-order scheme, it is necessary to also upgrade the implicit time stepping scheme to second order. We choose the implicit Trapezoidal rule in time with slope reconstruction in space. The question is whether the resulting mixed scheme, which we refer to as MUSCL-Trap, has second-order accuracy, too.
(Trapezoidal flux), depending on i, and h i = αh for i = 0 and h i = h otherwise. Choose ∆t such that 0 < λ = u∆t h ≤ 1, independent of α. Definition 2.1 (One step error) Denote bys n i the true cell averages on cell i at time t n . We define the one step error L(s n ,s n+1 ) i on cell i for the time step t n → t n+1 by Note that for standard schemes on uniform meshes the one step error is typically one order higher than the overall order of the scheme. One can show the following result [1,2]:

Numerical results
We support our theoretical findings by numerical results: we consider the model problem with α = 10 −5 , u = 1, and λ = 0.8 on I = [0, 1 + αh]. The small cell is located in the interval [0.5, 0.5 + αh]. The L 1 error has been normalized to adjust for the changing domain length. We use smooth initial data s 0 (x) = sin( 2πx 1+αh ) and choose T such that the test function is back to its original position. The results for the errors after 1 time step and at time T, respectively, are shown in Table 1: for the one step error we observe second-order convergence in the L ∞ norm due to limited accuracy of the cells around cell 0. Due to their number being O(1), this does not affect the third order convergence in the L 1 norm. At time T , we observe second-order convergence both in the L 1 and in the L ∞ norm, which is consistent with the statement of theorem 2.3.