Convergence analysis of PM-BDF2 method for quasiperiodic parabolic equations

TL;DR

This paper introduces a highly accurate and efficient numerical method combining projection and BDF2 techniques for solving complex quasiperiodic parabolic equations, with proven spectral accuracy in space and second-order in time.

Contribution

The paper presents a novel PM-BDF2 method with detailed convergence analysis and validation for quasiperiodic parabolic equations, enhancing numerical solution accuracy and efficiency.

Findings

Spectral accuracy in space achieved

Second-order accuracy in time demonstrated

Validated effectiveness in 1D and 2D cases

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.