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arXiv · 2412.19175

Convergence analysis of PM-BDF2 method for quasiperiodic parabolic equations

Kai Jiang·Meng Li·Juan Zhang·Lei Zhang
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Abstract

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.

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Kai Jiang, Meng Li, Juan Zhang, Lei Zhang. 2024-12-26. Convergence analysis of PM-BDF2 method for quasiperiodic parabolic equations. https://arxiv.org/abs/2412.19175

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