A direction splitting algorithm for incompressible flow in complex geometries

The paper proposes and examines the stability and convergence properties of an extension of the direction splitting methods towards parabolic and incompressible flow problems in complex, possibly time dependent geometries. Two possible spatial discretization are considered. One of them is based on a fictitious domain procedure and the other one is based on a grid adaptation around the domain boundary. The stability analysis of the discretization of the parabolic or Stokes problems in complex geometries is seriously complicated by the fact that the discrete second order spatial operators no longer commute if the geometry is different from a rectangle or parallelepiped. In this case, some of the classical direction splitting schemes loose stability in the three-dimensional case. We propose a modification of the Douglas scheme to avoid this problem. The numerical results suggest that the new technique is still unconditionally stable and retains the same convergence rate, in both time and space, as the classical Crank–Nicolson time discretization with a central difference discrete spatial operators. The stability in case of the unsteady Stokes problem is analyzed only if the parabolic part is not split. The analysis of the fully split scheme in this case remains an open problem.

► New direction splitting algorithms for incompressible ow in complex geometries.
► Stability and the consistency for parabolic problems and Navier–Stokes equations.
► Numerical results confirm that it is second order accurate in space and time.