Numerical solution of 2D Navier-Stokes equation discretized via boundary elements method and finite difference approximation

In this paper boundary elements method (BEM) is equipped with finite difference approximation (FDA) to solve two-dimensional Navier-Stokes (N-S) equation. The N-S equation is converted to a system of ordinary differential equations (ODEs) respect to time in streamfunction-vorticity formulation. The constant direct BEM and 9-point stencil FDA are utilized to handle spatial derivatives, and the final system of ODEs is solved via three numerical schemes, forward Euler, Runge-Kutta and Newton's methods to find a fast ODE solver. Numerical investigations presented in this article show Newton's method is faster than the others and it is able to solve the N-S equation for Reynolds number up to 40,000 when grid points are at most 141 x 141. Thanks to BEM and boundary conditions of lid-driven cavity flow, the final system of ODEs is stable also for higher Reynolds numbers. A new technique is proposed in this article which converts BEM's two-dimensional singular integrals to one-dimensional non-singular ones. The proposed technique reduces computational cost of BEM, significantly, when more accurate results are requested. Numerical experiments show the numerical results are fairly agree with that accurate ones available in the literature.