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Semi-Lagrangian Finite-Element Exterior Calculus for Incompressible Flows

by W. Tonnon and R. Hiptmair

(Report number 2023-07)

Abstract
We develop a mesh-based semi-Lagrangian discretization of the time-dependent incompressible Navier-Stokes equations with free boundary conditions recast as a non-linear transport problem for a momentum 1-form. A linearly implicit fully discrete version of the scheme enjoys excellent stability properties in the vanishing viscosity limit and is applicable to incviscid incompressible Euler flows. Conservation of energy is enforced separately.

Keywords: Euler equations, Navier-Stokes, Incompressible Flow, Lagrangian, discrete exterior calculus, Differential Forms, vanishing viscosity, deRham complex, structure-preserving, energy conservation, mass conservation, helicity conservation, mimetic

@Techreport{TH23_1044,
  author = {W. Tonnon and R. Hiptmair},
  title = {Semi-Lagrangian Finite-Element Exterior Calculus for Incompressible Flows},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2023-07},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2023/2023-07.pdf },
  year = {2023}
}

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