Reduction of the discretization stencil of direct forcing immersed boundary methods on rectangular cells: The ghost node shifting method

• Published in: Journal of computational physics 2018-07, Vol.364, p.18-48
• Main Authors: Picot, Joris, Glockner, Stéphane
• Format: Article
• Language: English
• Subjects: Aspect ratio; Boundary conditions; Cells; Computational fluid dynamics; Computational physics; Direct forcing; Dirichlet problem; Discretization; Discretization stencil; Finite difference method; Fluid flow; Immersed boundary method; Incompressible Navier–Stokes; Interpolation; Mathematical analysis; Navier-Stokes equations; Poisson distribution; Poisson problem; Simulation; Solvers
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2018.02.047

•Solving the Poisson problem with some immersed boundaries breaks the matrix stencil.•The discretization stencil of this problem unlimitly increases on rectangular grids.•These issues are solved with the proposed Ghost Shifting Method.•Immersed Boundaries on Navier–Stokes equations require an additional extrapolation.•Numerical simulations validate proposals with 2nd order convergence. We present an analytical study of discretization stencils for the Poisson problem and the incompressible Navier–Stokes problem when used with some direct forcing immersed boundary methods. This study uses, but is not limited to, second-order discretization and Ghost-Cell Finite-Difference methods. We show that the stencil size increases with the aspect ratio of rectangular cells, which is undesirable as it breaks assumptions of some linear system solvers. To circumvent this drawback, a modification of the Ghost-Cell Finite-Difference methods is proposed to reduce the size of the discretization stencil to the one observed for square cells, i.e. with an aspect ratio equal to one. Numerical results validate this proposed method in terms of accuracy and convergence, for the Poisson problem and both Dirichlet and Neumann boundary conditions. An improvement on error levels is also observed. In addition, we show that the application of the chosen Ghost-Cell Finite-Difference methods to the Navier–Stokes problem, discretized by a pressure-correction method, requires an additional interpolation step. This extra step is implemented and validated through well known test cases of the Navier–Stokes equations.