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An Explicit Implicit Scheme for Cut Cells in Embedded Boundary Meshes

by S. May and M. Berger

(Report number 2015-44)

Abstract
We present a new mixed explicit implicit time stepping scheme for solving the linear advection equation on a Cartesian cut cell mesh. Our scheme uses a standard second-order explicit scheme on Cartesian cells away from the embedded boundary. On cut cells, a second-order implicit scheme is used. 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. For one of them, which we call {\it flux bounding}, we can show a TVD result. We also discuss the solution of the resulting implicit systems. All components of the scheme have been kept simple enough to afford the extension of the scheme to three dimensions. Numerical results in one, two, and three dimensions indicate that the resulting scheme is second-order accurate in $L^1$ and between first- and second-order accurate along the embedded boundary.

Keywords: Cartesian cut cell method, finite volume scheme, embedded boundary grid, small cell problem, explicit implicit scheme

@Techreport{MB15_634,
  author = {S. May and M. Berger},
  title = {An Explicit Implicit Scheme for Cut Cells in Embedded Boundary Meshes},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2015-44},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2015/2015-44.pdf },
  year = {2015}
}

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