# About Convergence for Finite-difference Equations of Incompressible Fluid with Boundary Conditions by Woods Formulas

### Darkhan Akhmed-Zaki, Nargozy Danaev, Farida Amenova

#### 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.

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#### Paper Citation

#### in Harvard Style

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

#### in Bibtex Style

@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},

}

#### in EndNote Style

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