A time marching strategy for solving parabolic and elliptic equations with Neumann boundary conditions

Abstract

In the community of computational fluid dynamics, pressure Poisson equation with Neumann boundary condition is usually encountered when solving the incompressible Navier–Stokes equations in a segregated approach such as SIMPLE, PISO, and projection methods. To deal with Neumann boundary conditions more naturally and to retain high order spatial accuracy as well, a sixth-order accurate combined compact difference scheme developed on staggered grids (NSCCD6) is adopted to solve the parabolic and elliptic equations subject to Neumann boundary conditions. The staggered grid system is usually used when solving the incompressible Navier–Stokes equations. By adopting the combined compact difference concept, there is no need to discretize Neumann boundary conditions with one-sided discretization scheme which is of lower accuracy order. The conventional Crank–Nicolson scheme is applied in this study for temporal discretization. For two-dimensional cases, D’yakonov alternating direction implicit scheme is adopted. A newly proposed time step changing strategy is adopted to improve convergence rate when solving the steady state solutions of the parabolic equation. High accuracy order of the currently proposed NSCCD6 scheme for one- and two-dimensional cases are shown in this article.