# Highlight on immersed boundary method

The immersed boundary method (IBM) is a technique to represent boundaries of any shape over a Cartesian grid. Boundary conditions can be imposed at the immersed boundary, just like any other boundary. The method implemented in Notus strives to maintain good precision of the solutions by reaching a spatial convergence of second-order, while preserving performances on massively parallel applications.

Notus' IBM uses the Ghost-cell Finite-differences approach, in which available nodes from the outer side —a.k.a. ghost nodes— are used to sharply impose the boundary condition at the boundary; see Fig. 1. Figure 1: Définitions and notations of the computational domain, the immersed boundary, the mesh, and node types. The computational domain is split in inner and outer parts. Different symbols are used to distinguish inner (where solution is computed), ghost (used to ipose the boundary condition), and outer cell nodes (not used).

In  we propose the "Ghost Node Shifting Method" to reduce the discretization stencil of Direct forcing immersed boundary methods on rectangular cells. 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 (the linear method of Mittal , the direct method of Coco & Russo ). 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 (see figure 2 & 3). Figure 2: Examples of sets of interpolation nodes IP and IB on square cells. Notations of Fig. 1 applies. For the linear method, equation associated with the highlighted ghost node has a stencil of 2 as the probe point is farther than one cell width. For the direct method with the same discrete problem, equation associated with the highlighted ghost node has a stencil of 1 as the boundary point is closer that one cell width. Figure 3: Examples of sets of interpolation nodes IP and IB on rectangular cells a 4. Notations of Fig. 1 applies. For the linear method, equation associated with the highlighted ghost node has a stencil of 4, while for the direct method with the same discrete problem, equation associated with the highlighted ghost node has a stencil of 2. In the direct method, U(i,j) is not part of the set of interpolation nodes, and the boundary condition become ill posed.

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 (see figure 5). Figure 4: Examples of sets of interpolation nodes on rectangular cells. Notations of Fig. 1 applies. Using the Ghost Node Shifting Method on the same discrete problem than Fig. 3, equation associated with the highlighted ghost node has a reduced stencil of 2 for the linear method, and 1 for the direct method. In the direct method, U(i,j) is now part of the set of interpolation nodes. Figure 5: Norms L2 and Linf of the solution error for the Poisson problem with Neumann immersed boundary conditions and third-order interpolation. The non-shifted linear, shifted linear, and shifted direct methods are computed in each of the three mesh series. Although, the shifted linear method is not shown for the mesh series a 1, as it is equivalent to the non-shifted linear method.

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 (see figure 6).  Figure 6: Flows around cylinders. The streamlines are extracted from computations. Normalized pressure coefficient around the cylinders.

 J. Picot, S. Glockner, Discretization stencil reduction of Direct forcing immersed boundary methods on rectangular cells: the Ghost Node Shifting Method. Journal of Computational Physics, 364, pp18-48, 2018 (pdf).

 R. Mittal, H. Dong, M. Bozkurttas, F. M. Najjar, A. Vargas, A. von Loebbecke, A versatile sharp interface immersed boundary method for incompressible  ows with complex boundaries, Journal of Computational Physics 227 (10) (2008) 4825{4852, ISSN 0021-9991, doi:10.1016/j.jcp.2008.01.028.

 A. Coco, G. Russo, Finite-di erence ghost-point multigrid methods on Cartesian grids for elliptic problems in arbitrary domains, Journal of Computational Physics 241 (2013) 464{501, ISSN 0021-9991, doi:10.1016/j.jcp.2012.11.047.