The stability barrier of a class of Hermite-type discretizations of advection equations
Abstract
In this paper we fully categorize all hybrid-variable (HV) discretizations of linear advection equations according to their stability property and establish the stability barrier of these methods. In the HV discretization framework, we find numerical approximations to both cell-averaged solutions and nodal solutions and evolve them in time simultaneously. We prove that the HV methods have a similar stability barrier as the classical finite difference schemes, that the semi-discretized HV scheme is stable if and only if it uses a stencil such that the number of unknowns in the upwind direction is precisely that in the downwind direction plus one or two, with only one exception when the two numbers are three and zero, respectively. It is also proved that all central HV schemes are neutrally stable, in the sense that the L2-norm of the solution remain uniformly bounded at all times. The theoretical predictions are verify by extensive numerical tests.
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BibTeX
@misc{zeng2026stability,
title = {{The stability barrier of a class of Hermite-type discretizations of advection equations}},
author = {Xianyi Zeng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33201},
eprint = {2609.33201},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33201},
}
RIS
TY - UNPB
TI - The stability barrier of a class of Hermite-type discretizations of advection equations
AU - Xianyi Zeng
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33201
UR - https://arxiv.org/abs/2609.33201
AB - In this paper we fully categorize all hybrid-variable (HV) discretizations of linear advection equations according to their stability property and establish the stability barrier of these methods. In the HV discretization framework, we find numerical approximations to both cell-averaged solutions and nodal solutions and evolve them in time simultaneously. We prove that the HV methods have a similar stability barrier as the classical finite difference schemes, that the semi-discretized HV scheme is stable if and only if it uses a stencil such that the number of unknowns in the upwind direction is precisely that in the downwind direction plus one or two, with only one exception when the two numbers are three and zero, respectively. It is also proved that all central HV schemes are neutrally stable, in the sense that the L2-norm of the solution remain uniformly bounded at all times. The theoretical predictions are verify by extensive numerical tests.
ER -