High order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries

TL;DR

This paper addresses high-order accurate solutions of the Poisson equation on domains with curved boundaries by embedding the domain in a Cartesian grid and enforcing boundary conditions via ghost points. It develops and compares three fourth-order discretizations, two based on star-stencil and one on Mehrstellen (box) stencils, and employs level-set representations to handle complex geometries. The key findings are that one scheme is ill-conditioned, while the other two (Methods 2 and 3) achieve fourth-order accuracy for both the solution and its gradient, with Method 3 (Mehrstellen) delivering the best accuracy and practical performance, albeit requiring careful handling of the source term extrapolation. The results advocate the use of high-order ghost-point methods with multigrid solvers for efficient, accurate elliptic solves on irregular domains, with potential extensions to time-dependent problems and more general geometries.

Abstract

In this paper a fourth order finite difference ghost point method for the Poisson equation on regular Cartesian mesh is presented. The method can be considered the high order extension of the second ghost method introduced earlier by the authors. Three different discretizations are considered, which differ in the stencil that discretizes the Laplacian and the source term. It is shown that only two of them provide a stable method. The accuracy of such stable methods are numerically verified on several test problems.

Figures (11)

• Figure 1: Setup of problem \ref{['eq:modprob']}: the domain $\Omega$, defined by $\phi(x,y)\leq 0$, is embedded into a rectangular domain $R$. Mixed boundary conditions are imposed on the frontier $\Gamma$ of the domain. In each point of $\Gamma$ the unit normal pointing outside of the domain can be expressed as ${\mathbf{n}}=\nabla\phi/|\nabla\phi|$.
• Figure 2: Left panel: Shortley-Weller stencil around grid point $(i,j)$. Right panel: Ghost point discretization for the classic five-point discrete Laplacian. Boundary condition on node $G$ are assigned by imposing that polynomial that interpolated the solution on the (internal of ghost) nodes sarisfied the boundary conditions on point $B_G$, as in CocoRusso:Elliptic, or on intersection of the grid lines with the boundary (denoted by the asteriscs), as in Gibou:Ghost.
• Figure 3: Left: Stencil adopted in the discretization of the Laplace operator on a generic internal grid point $(x_i,y_j)$. For the star-stencil discretization \ref{['disc:star']}, the stencil is made by the purple grid points, while the dashed line surrounds the stencil of the box-stencil discretization \ref{['disc:box']}. Right: Star-shaped set of neighbour points $\mathcal{N}_{l,m}^{S,(1)}$ (red points) and $\mathcal{N}_{l,m}^{S,(2)}$ (blue points) for a generic ghost point $(x_l,y_m)$, as defined in Eq. \ref{['eq:starNeigh']}. The box-shaped set of neighbour points $\mathcal{N}_{l,m}^B$ (defined in Eq. \ref{['eq:boxNeigh']}) is surrounded by a dashed line. The set of neighbors of external grid points is adopted to identify the ghost points.
• Figure 4: Examples of stencils $\mathcal{S}_{l,m}^{(r_x,r_y)}$ adopted in the discretization of boundary conditions to determine the ghost point values. Left: stencils adopted in Method 1 and Method 3 . For Method 1 , the green point is a ghost point of $\Gamma^{S,(1)}_h$ and the purple point is a ghost point of $\Gamma^{S,(2)}_h$. The stencils are surrounded by the dashed lines of the same colour. Both stencils have $(r_x,r_y)=(1,1)$ in Eq. \ref{['eq:stencil']}. The respective boundary points are also represented. Right: stencils adopted in Method 2 . The stencil for a ghost point of $\Gamma^{S,(2)}_h$ (purple point) is enlarged, meaning that $(r_x,r_y)=(2,2)$ is adopted in Eq. \ref{['eq:stencil']}. The stencil is represented by the purple point (ghost point) and the purple circles. The stencil for the green ghost point, that is a ghost point of $\Gamma^{S,(2)}_h$, is unchanged with respect to the left plot.
• Figure 5: 1D grid nodes for quartic interpolation at $x^*$.

Theorems & Definitions (4)

• Remark 1
• Remark 2
• Remark 3
• Remark 4