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Energy stable boundary conditions for the nonlinear incompressible Navier-Stokes equations


Authors: Jan Nordström and Cristina La Cognata
Journal: Math. Comp.
MSC (2010): Primary 65M12, 65M06, 35M33, 76D05
DOI: https://doi.org/10.1090/mcom/3375
Published electronically: August 29, 2018
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Abstract: The nonlinear incompressible Navier-Stokes equations with different types of boundary conditions at far fields and solid walls is considered. Two different formulations of boundary conditions are derived using the energy method. Both formulations are implemented in both strong and weak form and lead to an estimate of the velocity field.

Equipped with energy bounding boundary conditions, the problem is approximated by using discrete derivative operators on summation-by-parts form and weak boundary and initial conditions. By mimicking the continuous analysis, the resulting semi-discrete as well as fully discrete scheme are shown to be provably stable, divergence free, and high-order accurate.


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Additional Information

Jan Nordström
Affiliation: Department of Mathematics, Computational Mathematics, Linköping University, Linköping, SE-581 83, Sweden
Email: jan.nordstrom@liu.se

Cristina La Cognata
Affiliation: Department of Mathematics, Computational Mathematics, Linköping University, Linköping, SE-581 83, Sweden
Email: cristina.la.cognata@liu.se

DOI: https://doi.org/10.1090/mcom/3375
Keywords: Navier--Stokes equations, incompressible, boundary conditions, energy estimate, stability, summation-by-parts, high-order accuracy, divergence free.
Received by editor(s): April 28, 2017
Received by editor(s) in revised form: December 31, 2017
Published electronically: August 29, 2018
Article copyright: © Copyright 2018 American Mathematical Society

American Mathematical Society