Accurately computing quasiperiodic parabolic equations within finite-size domains via modeling quasiperiodic boundary conditions

Abstract

Quasiperiodic systems exhibit long-range order without decay and are naturally posed on the whole space. However, in practical applications, computations are performed on finite domains, making the choice of boundary conditions that preserve the global quasiperiodic structure a key modeling challenge. In particular, conventional boundary conditions contain no information about the quasiperiodic field beyond the computational domain. Traditional periodic boundary conditions (PBCs) suffer from Diophantine errors due to the rational approximation of irrational numbers, limiting their accuracy. Motivated by this, we propose a class of quasiperiodic boundary conditions (QBCs) for quasiperiodic problems, which avoid the limitations caused by traditional Diophantine errors. By exploiting a homomorphism between a low-dimensional physical domain and a high-dimensional torus, QBCs effectively capture the long-range structure at the boundaries. To validate the proposed approach, we apply QBCs to solve quasiperiodic parabolic equations (QPEs) within finite-size domains and establish rigorous convergence results. Numerical experiments demonstrate that QBCs substantially reduce the influence of Diophantine errors. When employed to model finite-size QPEs and combined with suitable numerical discretizations, they enable accurate and efficient computations for both high- and low-regularity cases, while exhibiting improved convergence compared with PBCs.