A High-Accuracy Numerical Homogenization Method for Quasiperiodic Hamilton--Jacobi Equations
Abstract
In this work, we develop an accurate numerical homogenization method for computing effective Hamiltonians of quasiperiodic Hamilton--Jacobi equations (QHJEs) with convex Hamiltonians of the form H(x,p) = |p|k/k−f(x), k>1, where f is quasiperiodic. Computing effective Hamiltonians in the quasiperiodic setting requires solving QHJEs posed on the whole space. Their solutions generally possess neither translational symmetry nor decay and may exhibit low regularity. These features pose substantial challenges for numerical computation. To address these difficulties, we introduce a quasiperiodic boundary condition, which allows the original whole-space problem to be treated on a bounded domain while preserving quasiperiodicity at the boundary. We then propose an SL--FPR scheme that combines a semi-Lagrangian (SL) approximation with the finite points recovery (FPR) method and establish stability and error estimates for the resulting scheme. We also extend the quasiperiodic homogenization result from the quadratic case to general k>1 and apply the proposed method to accurately approximate the corresponding effective Hamiltonians. Numerical experiments illustrate the convergence and applicability of the method and validate the extended homogenization results.
Related
- Analytical and numerical periodic homogenization of Hamilton--Jacobi--Bellman and Isaacs equations in a W2,p Cordes framework
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- C^(sigma+alpha) regularity for fractional Hamilton-Jacobi equations, and rates of convergence in periodic homogenization
- Numerical homogenization of a linear multiscale conservation law
BibTeX
@misc{jiang2026highaccuracy,
title = {{A High-Accuracy Numerical Homogenization Method for Quasiperiodic Hamilton--Jacobi Equations}},
author = {Kai Jiang and Meng Li and Juan Zhang and Lei Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.24313},
eprint = {2608.24313},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.24313},
}
RIS
TY - UNPB
TI - A High-Accuracy Numerical Homogenization Method for Quasiperiodic Hamilton--Jacobi Equations
AU - Kai Jiang
AU - Meng Li
AU - Juan Zhang
AU - Lei Zhang
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2608.24313
UR - https://arxiv.org/abs/2608.24313
AB - In this work, we develop an accurate numerical homogenization method for computing effective Hamiltonians of quasiperiodic Hamilton--Jacobi equations (QHJEs) with convex Hamiltonians of the form H(x,p) = |p|^k/k−f(x), k>1, where f is quasiperiodic. Computing effective Hamiltonians in the quasiperiodic setting requires solving QHJEs posed on the whole space. Their solutions generally possess neither translational symmetry nor decay and may exhibit low regularity. These features pose substantial challenges for numerical computation. To address these difficulties, we introduce a quasiperiodic boundary condition, which allows the original whole-space problem to be treated on a bounded domain while preserving quasiperiodicity at the boundary. We then propose an SL--FPR scheme that combines a semi-Lagrangian (SL) approximation with the finite points recovery (FPR) method and establish stability and error estimates for the resulting scheme. We also extend the quasiperiodic homogenization result from the quadratic case to general k>1 and apply the proposed method to accurately approximate the corresponding effective Hamiltonians. Numerical experiments illustrate the convergence and applicability of the method and validate the extended homogenization results.
ER -