Constructing stable, high-order finite-difference operators on point clouds over complex geometries
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| Format: | Preprint |
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2024
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| _version_ | 1866912206282555392 |
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| author | Hicken, Jason Yan, Ge Kaur, Sharanjeet |
| author_facet | Hicken, Jason Yan, Ge Kaur, Sharanjeet |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_00809 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Constructing stable, high-order finite-difference operators on point clouds over complex geometries Hicken, Jason Yan, Ge Kaur, Sharanjeet Numerical Analysis 65M06, 65M70, 65M12 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. |
| title | Constructing stable, high-order finite-difference operators on point clouds over complex geometries |
| topic | Numerical Analysis 65M06, 65M70, 65M12 |
| url | https://arxiv.org/abs/2409.00809 |