发表机构
Xiangtan University; Wuhan University（湘潭大学; 武汉大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对有限域内准周期抛物型方程计算的丢番图误差问题，本文提出准周期边界条件，可有效降低误差并实现精确高效的计算，性能优于传统周期边界条件。
AI 中文摘要
准周期系统呈现无衰减的长程有序性，天然定义于整个空间。但在实际应用中，计算需在有限域内进行，因此选择能保留全局准周期结构的边界条件成为关键建模挑战。传统边界条件不包含计算域外准周期场的信息，传统周期边界条件（PBCs）因无理数的有理近似会产生丢番图误差，限制了计算精度。受此启发，本文针对准周期问题提出一类准周期边界条件（QBCs），可避免传统丢番图误差带来的局限。通过利用低维物理域与高维环面之间的同态，QBCs能有效在边界处捕捉长程结构。为验证所提方法，将QBCs应用于求解有限尺寸域内的准周期抛物型方程（QPEs），并建立严格的收敛结果。数值实验表明，QBCs大幅降低了丢番图误差的影响；当用于建模有限尺寸QPEs并结合合适的数值离散化时，其在高正则性和低正则性情形下均能实现精确高效的计算，且相比PBCs展现出更优的收敛性。
英文摘要
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.
