Constructing stable, high-order finite-difference operators on point clouds over complex geometries

TL;DR

This work develops high-order diagonal-norm SBP operators on point clouds for complex geometries by leveraging a automatically generated background Cartesian cut-cell mesh. Operators are built from cell-based degenerate SBP pairs assembled into a global operator, with a linear inequality used to enforce a positive-definite diagonal mass matrix, ensuring stability for hyperbolic problems. The approach achieves up to fifth-order design accuracy, demonstrates stability and accuracy for linear advection, and provides insights into quadrature properties and computational costs, while highlighting that point-cloud SBP is less sparse than tensor-product counterparts. The methodology enables stable, high-order discretizations on complex domains with automatic preprocessing, offering a practical path toward efficient, geometry-flexible simulations and potential future integration with RBF-FD and parallel implementations.

Abstract

High-order difference operators with the summation-by-parts (SBP) property can be used to build stable discretizations of hyperbolic conservation laws; however, most high-order SBP operators require a conforming, high-order mesh for the domain of interest. To circumvent this requirement, we present an algorithm for building high-order, diagonal-norm, first-derivative SBP operators on point clouds over level-set geometries. The algorithm is not mesh-free, since it uses a Cartesian cut-cell mesh to define the sparsity pattern of the operators and to provide intermediate quadrature rules; however, the mesh is generated automatically and can be discarded once the SBP operators have been constructed. Using this temporary mesh, we construct local, cell-based SBP difference operators that are assembled into global SBP operators. We identify conditions for the existence of a positive-definite diagonal mass matrix, and we compute the diagonal norm by solving a sparse system of linear inequalities using an interior-point algorithm. We also describe an artificial dissipation operator that complements the first-derivative operators when solving hyperbolic problems, although the dissipation is not required for stability. The numerical results confirm the conditions under which a diagonal norm exists and study the distribution of the norm's entries. In addition, the results verify the accuracy and stability of the point-cloud SBP operators using the linear advection equation.

Figures (13)

• Figure 1: Example of a domain (Fig. \ref{['fig:ex_domain']}) and corresponding background mesh (Fig. \ref{['fig:ex_mesh']}) and node set (Fig. \ref{['fig:ex_points']}).
• Figure 2: Illustration of a cell (dark gray) with its stencil (red circles). The light gray squares and partial squares in the lower left indicate the immersed region where $\phi(\bm{x}) < 0$. The dashed circle provides a reference length to verify that only the closest $N^{\mathsf{c}} = 5$ nodes are included in the stencil. The blue squares represent quadrature nodes for integration over the (cut) cell.
• Figure 3: Illustration of the nodes and quadrature points involved in the interpolation operators $\mathsf{R}^{\mathsf{f},-}$ and $\mathsf{R}^{\mathsf{f},+}$ for a face $\mathsf{f}$ between two adjacent cells $\mathsf{c}^{-}$ and $\mathsf{c}^{+}$ (dark gray squares). The stencil for the left cell is depicted with large white circles while the stencil for the right cell uses red circles; note that five nodes are shared by the stencils.
• Figure 4: Airfoil geometry $\Omega_{\text{foil}}$ and an example node distribution used for the accuracy studies.
• Figure 5: Quadrature-weight study for the annulus with quasi-uniform nodes. The left and center columns show the weights before and after solving the linear inequality, respectively. The histograms in the right column show the distribution of the entries.

Theorems & Definitions (19)

• Definition 1: Degenerate SBP pair
• Definition 2: Diagonal-Norm SBP Operator
• Theorem 1: SBP quadrature
• Theorem 2: SBP existence
• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Theorem 3
• proof