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

TL;DR

This paper introduces an algorithm to construct high-order, stable finite-difference operators on point clouds over complex geometries, enabling accurate hyperbolic PDE solutions without requiring conforming meshes.

Contribution

The authors develop a novel method to build SBP operators on point clouds using a temporary Cartesian cut-cell mesh, allowing stable discretizations on complex geometries without conforming meshes.

Findings

Successfully constructed SBP operators with positive-definite diagonal mass matrices.

Verified the accuracy and stability of the operators on the linear advection equation.

Analyzed the distribution of the diagonal norm entries for the SBP operators.

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.