Conservative polytopal mimetic discretization of the incompressible Navier–Stokes equations

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Abstract

We discretize the incompressible Navier–Stokes equations on a polytopal mesh by using mimetic reconstruction operators. The resulting method conserves discrete mass, momentum, and kinetic energy in the inviscid limit, and determines the vorticity such that the global vorticity is consistent with the boundary conditions. To do this we introduce a dual mesh and show how the dual mesh can be completed to a cell-complex. We present existing mimetic reconstruction operators in a new symmetric way applicable to arbitrary dimension, use these to interpolate between primal and dual mesh and derive properties of these operators. Finally, we test both 2- and 3-dimensional versions of the method on a variety of complicated meshes to show its wide applicability. We numerically test the convergence of the method and verify the derived conservation statements.

Original languageEnglish
Pages (from-to)443-473
Number of pages31
JournalJournal of Computational and Applied Mathematics
Volume340
Issue number1 October 2018
DOIs
Publication statusPublished - 1 Oct 2018

Keywords

  • Cell-complex
  • Exact discrete conservation
  • Exterior calculus
  • Incompressible Navier–Stokes equations
  • Mimetic discretization
  • Primal and dual meshes
  • Incompressible Navier-Stokes equations

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