Numerical Simulations of Stokes Flow by the Iterations of Boundary Conditions and Finite Difference Methods

Abstract

In this study the iteration of boundary conditions method (Chizhonkov and Kargin, 2006) is used together with the well known Finite difference numerical method to solve the Stokes problem over a rectangular domain as well as in irregular domain. The iteration of boundary conditions method has been applied to the Stokes problem in a rectangular domain, 􀀀

2 <x<

2 , 􀀀 d 2 < y < d 2 , by the above mentioned researchers. Our main task here is to validate the results of the approximate methods by this analytical method in case of the rectangular domain and extend that to the case of irregular domain.The (Chizhonkov and Kargin, 2006) algorithm is typically the best choice for validation purposes because of its high accuracy. It is known in literature that increasing the parameter d, which represents the ratio of the sides, leads to slow down in convergence of the approximate methods like the conjugate Gradients of Uzawa (Kobelkov and Olshanskii, 2000). It is therefore important that an algorithm that converges uniformly with respect to the parameter d is considered. The (Chizhonkov and Kargin, 2006) algorithm is typical of such an algorithm, and hence our choice of the method in this work. In this project the non-homogeneous Stokes problem is transformed into a homogeneous Stokes problem and the resulting problem is then decomposed into two sub problems that are solvable by the eigenfunction expansion method. Once all necessary coefficients of the generalised Fourier series are known and the functions describing the boundary conditions are prescribed and represented in terms of the Fourier series, we then proceed to formulate the iteration of boundary conditions numerical algorithm. Finally we develop a numerical scheme, using the finite difference methods, for solving the problem in both rectangular and irregular domains. Coding of the numerical algorithm is done using MATLAB 9.0,R2016a programming language, and implemented by the author. The results of the two methods in both cases of boundary conditions are then compared for validation of our purely numerical results.