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.