On a class of supraconvergent Hermite-type methods for advection-diffusion equations
Abstract
We develop and analyze a class of Hermite-type methods for linear advection-diffusion equations in the case of one-dimensional periodic domains. The methods evolve both nodal values and cell averages and are therefore referred to as hybrid-variable (HV) methods. Using Hermite interpolation theory, we construct the HV methods to arbitrary order of accuracy, and prove that the method is supraconvergent in the sense that the spatial order of accuracy can be up to two orders larger than the local truncation error. In the second part, we prove all central HV methods for linear diffusion equations and linear advection-diffusion equations are stable at the semi-discretized level. Next, we prove the convergence of all stable HV schemes and provide error estimate in the discrete L2-norm of the semi-discretized solution, which extends the supraconvergence from a single mode to all sufficiently smooth initial data. The theoretical predictions on the accuracy and stability of the methods are verified by extensive numerical tests.
Related
- The stability barrier of a class of Hermite-type discretizations of advection equations
- Finite difference methods for three kinds of reaction-diffusion equations with free boundaries
- A New Modifications of the Milstein and Euler Methods and the Taylor-Ito Expansion for Ito Stochastic Differential Equations with Drift and Diffusion Coefficients Having Non-Differentiable or Singular Time-Dependent Multipliers
- Sign-switched antithetic coupling in gradient random-walk methods for convection-diffusion equations
- Finite element exterior calculus for spectra and pseudospectra of advection-diffusion of differential forms
BibTeX
@misc{zeng2026class,
title = {{On a class of supraconvergent Hermite-type methods for advection-diffusion equations}},
author = {Xianyi Zeng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.14545},
eprint = {2609.14545},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14545},
}
RIS
TY - UNPB
TI - On a class of supraconvergent Hermite-type methods for advection-diffusion equations
AU - Xianyi Zeng
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2609.14545
UR - https://arxiv.org/abs/2609.14545
AB - We develop and analyze a class of Hermite-type methods for linear advection-diffusion equations in the case of one-dimensional periodic domains. The methods evolve both nodal values and cell averages and are therefore referred to as hybrid-variable (HV) methods. Using Hermite interpolation theory, we construct the HV methods to arbitrary order of accuracy, and prove that the method is supraconvergent in the sense that the spatial order of accuracy can be up to two orders larger than the local truncation error. In the second part, we prove all central HV methods for linear diffusion equations and linear advection-diffusion equations are stable at the semi-discretized level. Next, we prove the convergence of all stable HV schemes and provide error estimate in the discrete L^2-norm of the semi-discretized solution, which extends the supraconvergence from a single mode to all sufficiently smooth initial data. The theoretical predictions on the accuracy and stability of the methods are verified by extensive numerical tests.
ER -