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About Convergence for Finite-difference Equations of Incompressible Fluid with Boundary Conditions by Woods FormulasTopics: Dynamical Systems Models and Methods; Fluid Dynamics; Non-Linear Systems

Keyword(s):Two-dimensional System of the Navier-stokes Equations for an Incompressible Fluid, Linear Stokes Differential Problem, Method of a Priori Estimates, Stability, Convergence, Iterative Algorithm.

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Ontology
Subjects/Areas/Topics:Complex Systems Modeling and Simulation
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Dynamical Systems Models and Methods
;
Fluid Dynamics
;
Formal Methods
;
Non-Linear Systems
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Simulation and Modeling

Abstract: In this paper, mathematical aspects of stability, convergence and numerical implementation of two-dimensional
differential problem for incompressible fluid equations in “stream function, vorticity” variables
defined on a symmetrical template of finite-difference grid studied by method of a priori estimates are considered.
Approximate boundary conditions for the vorticity are chosen in the form of Woods formula. In case
of a linear Stokes problem, it is shown that the numerical solution of the difference problem converges to the
solution of the differential problem with second order accuracy and two algorithms of numerical implementation,
for which the rates of convergence obtained, are considered. In the case of non-linear Navier-Stokes
equations, estimates of the convergence of a solution of the difference problem to the solution of the differential
problem, as well as estimation of the convergence of a considered iterative algorithm with the assumption
that the condition is equivalent to the condition of uniqueness of nonlinear difference problem are obtained.(More)

In this paper, mathematical aspects of stability, convergence and numerical implementation of two-dimensional differential problem for incompressible fluid equations in “stream function, vorticity” variables defined on a symmetrical template of finite-difference grid studied by method of a priori estimates are considered. Approximate boundary conditions for the vorticity are chosen in the form of Woods formula. In case of a linear Stokes problem, it is shown that the numerical solution of the difference problem converges to the solution of the differential problem with second order accuracy and two algorithms of numerical implementation, for which the rates of convergence obtained, are considered. In the case of non-linear Navier-Stokes equations, estimates of the convergence of a solution of the difference problem to the solution of the differential problem, as well as estimation of the convergence of a considered iterative algorithm with the assumption that the condition is equivalent to the condition of uniqueness of nonlinear difference problem are obtained.

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Akhmed-Zaki, D.; Danaev, N. and Amenova, F. (2014). About Convergence for Finite-difference Equations of Incompressible Fluid with Boundary Conditions by Woods Formulas.In Proceedings of the 4th International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: SIMULTECH, ISBN 978-989-758-038-3, pages 413-420. DOI: 10.5220/0005034204130420

@conference{simultech14, author={Darkhan Akhmed{-}Zaki. and Nargozy Danaev. and Farida Amenova.}, title={About Convergence for Finite-difference Equations of Incompressible Fluid with Boundary Conditions by Woods Formulas}, booktitle={Proceedings of the 4th International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: SIMULTECH,}, year={2014}, pages={413-420}, publisher={SciTePress}, organization={INSTICC}, doi={10.5220/0005034204130420}, isbn={978-989-758-038-3}, }

TY - CONF

JO - Proceedings of the 4th International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: SIMULTECH, TI - About Convergence for Finite-difference Equations of Incompressible Fluid with Boundary Conditions by Woods Formulas SN - 978-989-758-038-3 AU - Akhmed-Zaki, D. AU - Danaev, N. AU - Amenova, F. PY - 2014 SP - 413 EP - 420 DO - 10.5220/0005034204130420