PreprintarXiv licencearXiv
Spectral Reality for Certain Fourth-Order Nonsymmetric Finite-Difference Laplacians
Abstract
Finite-difference discretizations of Laplace operators yield discrete Laplacians whose spectral properties are closely tied to the stability, convergence, and physical fidelity of numerical solvers. While symmetric discretizations are well understood, many high-order finite-difference schemes produce nonsymmetric matrices for which rigorous spectral analysis is overlooked. Surprisingly, we prove that two fourth-order schemes with Dirichlet boundary conditions yield nonsymmetric discrete Laplacians whose spectra are purely real and strictly positive. Computer-assisted computations show that this property does not hold for higher-order extensions of these schemes in general.
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BibTeX
@misc{feng2026spectral,
title = {{Spectral Reality for Certain Fourth-Order Nonsymmetric Finite-Difference Laplacians}},
author = {Yizhe Feng and Weiguo Gao and Meiyue Shao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.32410},
eprint = {2609.32410},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32410},
}
RIS
TY - UNPB
TI - Spectral Reality for Certain Fourth-Order Nonsymmetric Finite-Difference Laplacians
AU - Yizhe Feng
AU - Weiguo Gao
AU - Meiyue Shao
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2609.32410
UR - https://arxiv.org/abs/2609.32410
AB - Finite-difference discretizations of Laplace operators yield discrete Laplacians whose spectral properties are closely tied to the stability, convergence, and physical fidelity of numerical solvers. While symmetric discretizations are well understood, many high-order finite-difference schemes produce nonsymmetric matrices for which rigorous spectral analysis is overlooked. Surprisingly, we prove that two fourth-order schemes with Dirichlet boundary conditions yield nonsymmetric discrete Laplacians whose spectra are purely real and strictly positive. Computer-assisted computations show that this property does not hold for higher-order extensions of these schemes in general.
ER -