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Stability and convergence of efficient Navier‐Stokes solvers via a commutator estimate
Author(s) -
Liu JianGuo,
Liu Jie,
Pego Robert L.
Publication year - 2007
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20178
Subject(s) - mathematics , discretization , bounded function , commutator , stokes flow , mathematical analysis , mathematical proof , navier–stokes equations , helmholtz equation , compressibility , convergence (economics) , boundary value problem , pure mathematics , algebra over a field , geometry , flow (mathematics) , lie conformal algebra , economic growth , engineering , economics , aerospace engineering
For strong solutions of the incompressible Navier‐Stokes equations in bounded domains with velocity specified at the boundary, we establish the unconditional stability and convergence of discretization schemes that decouple the updates of pressure and velocity through explicit time stepping for pressure. These schemes require no solution of stationary Stokes systems, nor any compatibility between velocity and pressure spaces to ensure an inf‐sup condition, and are representative of a class of highly efficient computational methods that have recently emerged. The proofs are simple, based upon a new, sharp estimate for the commutator of the Laplacian and Helmholtz projection operators. This allows us to treat an unconstrained formulation of the Navier‐Stokes equations as a perturbed diffusion equation. © 2007 Wiley Periodicals, Inc.