A fourth-order accurate compact difference scheme for solving the three-dimensional Poisson equation with arbitrary boundaries

TL;DR

The paper addresses solving the 3D Poisson equation with immersed boundaries on nonuniform grids by introducing a fourth-order compact finite-difference scheme that preserves sharp interfaces without jump corrections. A sharp immersed interface method modifies 27-point stencils near the boundary and a second-order seven-point FD preconditioner enables a preconditioned BiCGSTAB solver whose efficiency is independent of geometry. Spectral analysis confirms stability under immersed boundaries, and extensive numerical experiments across simple and complex geometries demonstrate robust fourth-order accuracy in the maximum norm with minimal additional FLOPs. The approach offers a practical, high-accuracy, geometry-robust solver for 3D Poisson problems in complex domains.

Abstract

This paper presents an efficient high-order sharp-interface method for solving the three-dimensional (3D) Poisson equation with Dirichlet boundary conditions on a nonuniform Cartesian grid with irregular domain boundaries. The new approach is based on the combination of the fourth-order compact finite difference scheme and the preconditioned stabilized biconjugate-gradient (BiCGSTAB) method. Contrary to the original immersed interface method by LeVeque and Li [1], the new method does not require jump corrections, instead, the (regular) compact finite difference stencil is adjusted at the irregular grid points (in the vicinity of the interfaces of the immersed bodies) to obtain a solution that is sharp across the interface while keeping the fourth-order global accuracy. The contribution of the present work is the design of a fourth-order-accurate 3D Poisson solver whose accuracy and efficiency does not deteriorate in the presence of an immersed boundary. This is attributed to (i) the modification of the discrete operators near immersed boundaries does not lead to a wide grid stencil due to the compact nature of the discretization and (ii) a preconditioning technique whose efficiency and cost are independent of the complexity of the geometry and the presence or not of an immersed boundary. The accuracy and computational efficiency of the proposed algorithm is demonstrated and validated over a range of problems including smooth and irregular boundaries. The test cases show that the new method is fourth-order accurate in the maximum norm whether an immersed boundary is present or not, on uniform or nonuniform grids. Furthermore, the efficiency of the preconditioned BiCGSTAB is demonstrated with regard to convergence rate and `extra' floating-point operation ($FLOP_{extra}$) which is due to the presence of immersed boundaries.

Figures (12)

• Figure 1: Representation of the different type of grid points.
• Figure 2: (a) Intersection of 27-point compact FD stencil at irregular grid point ($i,j,k$) with an immersed boundary. Circles are additional grid points used in the modified finite difference operators to maintain the formal fourth-order accuracy. IBI points are numbered from 1 to 9, i.e. $IBI_1,~IBI_2, \cdots,~IBI_9$. Modified finite difference (MFD) operators employed in the $z-$direction along the plane $x=x_{i+1}$ at $y=y_{J}$ with $J=j$ & $J=j+1$ (b) and $y=y_{j-1}$ (c). MFD operators used in the $x-$direction corresponding to $IBI_1-IBI_5$ (d), and MFD operators employed in the $y$-direction along the plane $x=x_{i+1}$ at $z=z_k$ (e). Grid points used on the RHS of the MFD operators are marked with an up-arrow sign. Grid points enclosed by rectangles are the points used in the LHS of the MFD operators.
• Figure 3: (a) Schematic of double intersections with immersed boundary for irregular grid point centered at ($i,j,k$). Points of intersection with the immersed boundary are numbered from 1 to 9. (b) Modified finite difference operators employed in the $z-$direction along the plane $x=x_{i+1}$ at $y=y_{J}$ with $J=j-1$, $J=j$ and $J=j+1$. For the descriptions regarding the colored rectangles and marked points, see caption of Fig. \ref{['fig:irr_grid_point']}.
• Figure 4: Eigenvalue spectra of the discretization matrices obtained for a uniform grid with $21^3$ points. (a) Simple domain and (b) domain with an immersed sphere with radius of 0.2. Inserts are close-up views of the spectra near the origin.
• Figure 5: (a) Finite difference seven-point stencil at regular grid point centered at ($i,j,k$). (b) The intersection of seven-point stencil with an immersed boundary.