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Time delay and finite differences for the non‐stationary non‐linear Navier–Stokes equations
Author(s) -
Varnhorn Werner
Publication year - 1992
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1670150204
Subject(s) - mathematics , sobolev space , bounded function , navier–stokes equations , limit (mathematics) , mathematical analysis , sequence (biology) , convergence (economics) , energy (signal processing) , compressibility , biology , economic growth , engineering , economics , genetics , aerospace engineering , statistics
In the present paper we use a time delay ϵ > 0 for an energy conserving approximation of the non‐linear term of the non‐stationary Navier–Stokes equations. We prove that the corresponding initial‐value problem (N ϵ ) in smoothly bounded domains G ⊆ ℝ 3 is well‐posed. We study a semidiscretized difference scheme for (N ϵ ) and prove convergence to optimal order in the Sobolev space H 2 ( G ). Passing to the limit ϵ→0 we show that the sequence of stabilized solutions has an accumulation point such that it solves the Navier–Stokes problem (N o ) in a weak sense (Hopf).