An Explicit Implicit Scheme for Cut Cells in Embedded Boundary Meshes

Sandra May, Marsha Berger

Research output: Contribution to journalArticlepeer-review


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 L1 and between first- and second-order accurate along the embedded boundary in two and three dimensions.

Original languageEnglish (US)
Pages (from-to)919-943
Number of pages25
JournalJournal of Scientific Computing
Issue number3
StatePublished - Jun 1 2017


  • Cartesian cut cell method
  • Embedded boundary grid
  • Explicit implicit scheme
  • Finite volume scheme
  • Small cell problem

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science
  • Numerical Analysis
  • General Engineering
  • Computational Theory and Mathematics
  • Computational Mathematics
  • Applied Mathematics


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