A posteriori modeling error estimates for the assumption of perfect incompressibility in the Navier-Stokes equation

J.L. Fischer, SIAM Journal on Numerical Analysis 53 (2015) 2178–2205.

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Journal Article | Published | English
Abstract
We derive a posteriori estimates for the modeling error caused by the assumption of perfect incompressibility in the incompressible Navier-Stokes equation: Real fluids are never perfectly incompressible but always feature at least some low amount of compressibility. Thus, their behavior is described by the compressible Navier-Stokes equation, the pressure being a steep function of the density. We rigorously estimate the difference between an approximate solution to the incompressible Navier-Stokes equation and any weak solution to the compressible Navier-Stokes equation in the sense of Lions (without assuming any additional regularity of solutions). Heuristics and numerical results suggest that our error estimates are of optimal order in the case of "well-behaved" flows and divergence-free approximations of the velocity field. Thus, we expect our estimates to justify the idealization of fluids as perfectly incompressible also in practical situations.
Publishing Year
Date Published
2015-01-01
Journal Title
SIAM Journal on Numerical Analysis
Acknowledgement
The research of the author was supported by the Lithuanian-Swiss cooperation program under the project agreement CH-SMM-01/0.
Volume
53
Issue
5
Page
2178 - 2205
IST-REx-ID

Cite this

Fischer JL. A posteriori modeling error estimates for the assumption of perfect incompressibility in the Navier-Stokes equation. SIAM Journal on Numerical Analysis. 2015;53(5):2178-2205. doi:10.1137/140966654
Fischer, J. L. (2015). A posteriori modeling error estimates for the assumption of perfect incompressibility in the Navier-Stokes equation. SIAM Journal on Numerical Analysis, 53(5), 2178–2205. https://doi.org/10.1137/140966654
Fischer, Julian L. “A Posteriori Modeling Error Estimates for the Assumption of Perfect Incompressibility in the Navier-Stokes Equation.” SIAM Journal on Numerical Analysis 53, no. 5 (2015): 2178–2205. https://doi.org/10.1137/140966654.
J. L. Fischer, “A posteriori modeling error estimates for the assumption of perfect incompressibility in the Navier-Stokes equation,” SIAM Journal on Numerical Analysis, vol. 53, no. 5, pp. 2178–2205, 2015.
Fischer JL. 2015. A posteriori modeling error estimates for the assumption of perfect incompressibility in the Navier-Stokes equation. SIAM Journal on Numerical Analysis. 53(5), 2178–2205.
Fischer, Julian L. “A Posteriori Modeling Error Estimates for the Assumption of Perfect Incompressibility in the Navier-Stokes Equation.” SIAM Journal on Numerical Analysis, vol. 53, no. 5, Society for Industrial and Applied Mathematics , 2015, pp. 2178–205, doi:10.1137/140966654.

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