Accurately computing quasiperiodic parabolic equations within finite-size domains via modeling quasiperiodic boundary conditions
Abstract
Quasiperiodic parabolic equations (QPEs) describe various physical processes in systems with long-range order without decay, such as heat conduction, diffusion, and transport, and are naturally posed on the whole space. However, practical computations could be performed on finite domains, where preserving the global quasiperiodic structure through appropriate boundary conditions remains a key challenge. Traditional periodic boundary conditions (PBCs) cannot capture quasiperiodic information beyond the computational domain and suffer from Diophantine errors caused by rational approximations of irrational numbers. To address this issue, we propose a class of quasiperiodic boundary conditions (QBCs) based on a homomorphism between the physical domain and a high-dimensional torus, enabling the long-range quasiperiodic structure to be captured at the boundaries. We further establish rigorous convergence results for numerical discretizations of finite-size QPEs with QBCs. Numerical experiments demonstrate that QBCs avoid the influence of Diophantine errors and enable accurate and efficient computations for both high- and low-regularity cases, with improved convergence over PBCs.
Related
- Simultaneous Reconstructions of (x,t) -Dependent Coefficients and Initial Conditions of Parabolic Equations
- A High-Accuracy Numerical Homogenization Method for Quasiperiodic Hamilton--Jacobi Equations
- A doubly nonlinear parabolic problem with variable exponents, homogeneous Neumann boundary conditions and generalized logistic source
- An Immersed Interface Method for Parabolic Interface Problems with Nonlinear Jump Conditions
- Harnack-type inequalities for nonlocal doubly nonlinear parabolic equations
BibTeX
@misc{han2026accurately,
title = {{Accurately computing quasiperiodic parabolic equations within finite-size domains via modeling quasiperiodic boundary conditions}},
author = {Xiaofang Han and Kai Jiang and Meng Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.22241},
eprint = {2608.22241},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.22241},
}
RIS
TY - UNPB
TI - Accurately computing quasiperiodic parabolic equations within finite-size domains via modeling quasiperiodic boundary conditions
AU - Xiaofang Han
AU - Kai Jiang
AU - Meng Li
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2608.22241
UR - https://arxiv.org/abs/2608.22241
AB - Quasiperiodic parabolic equations (QPEs) describe various physical processes in systems with long-range order without decay, such as heat conduction, diffusion, and transport, and are naturally posed on the whole space. However, practical computations could be performed on finite domains, where preserving the global quasiperiodic structure through appropriate boundary conditions remains a key challenge. Traditional periodic boundary conditions (PBCs) cannot capture quasiperiodic information beyond the computational domain and suffer from Diophantine errors caused by rational approximations of irrational numbers. To address this issue, we propose a class of quasiperiodic boundary conditions (QBCs) based on a homomorphism between the physical domain and a high-dimensional torus, enabling the long-range quasiperiodic structure to be captured at the boundaries. We further establish rigorous convergence results for numerical discretizations of finite-size QPEs with QBCs. Numerical experiments demonstrate that QBCs avoid the influence of Diophantine errors and enable accurate and efficient computations for both high- and low-regularity cases, with improved convergence over PBCs.
ER -