Solution of the incompressible Navier-Stokes equations by the method of lines

The numerical method of lines (NUMOL) is a numerical technique used to solve efficiently partial differential equations. In this paper, the NUMOL is applied to the solution of the two-dimensional unsteady Navier-Stokes equations for incompressible laminar flows in Cartesian coordinates. The Navier-Stokes equations are first discretized (in space) on a staggered grid as in the Marker and Cell scheme. The discretized Navier-Stokes equations form an index 2 system of differential algebraic equations, which are afterwards reduced to a system of ordinary differential equations (ODEs), using the discretized form of the continuity equation. The pressure field is computed solving a discrete pressure Poisson equation. Finally, the resulting ODEs are solved using the backward differentiation formulas. The proposed method is illustrated with Dirichlet boundary conditions through applications to the driven cavity flow and to the backward facing step flow. Copyright (c) 2015 John Wiley & Sons, Ltd.