Numerically solving parabolic equations with quasiperiodic coefficients is a significant challenge due to the potential formation of space-filling quasiperiodic structures that lack translational symmetry or decay. In this paper, we introduce a highly accurate numerical method for solving time-dependent quasiperiodic parabolic equations. We discretize the spatial variables using the projection method (PM) and the time variable with the second-order backward differentiation formula (BDF2). We provide a complexity analysis for the resulting PM-BDF2 method. Furthermore, we conduct a detailed convergence analysis, demonstrating that the proposed method exhibits spectral accuracy in space and second-order accuracy in time. Numerical results in both one and two dimensions validate these convergence results, highlighting the PM-BDF2 method as a highly efficient algorithm for addressing quasiperiodic parabolic equations.

翻译：数值求解具有拟周期系数的抛物方程是一项重大挑战，因为可能形成缺乏平移对称性或衰减性的空间填充拟周期结构。本文提出了一种高精度数值方法用于求解含时拟周期抛物方程。我们采用投影方法（PM）对空间变量进行离散化，并利用二阶向后差分公式（BDF2）对时间变量进行离散。我们对由此得到的PM-BDF2方法进行了复杂度分析。此外，我们开展了详细的收敛性分析，证明所提方法在空间上具有谱精度，在时间上具有二阶精度。一维和二维的数值结果验证了这些收敛性结论，表明PM-BDF2方法是处理拟周期抛物方程的高效算法。