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Convergence analysis of PM-BDF2 method for quasiperiodic parabolic equations

Kai Jiang, Meng Li, Juan Zhang, Lei Zhang

TL;DR

This work tackles the numerical solution of time-dependent quasiperiodic parabolic equations with space-quasiperiodic coefficients by marrying the projection method (PM) in space with the second-order backward differentiation formula (BDF2) in time. The authors develop a PM-BDF2 discretization, analyze its computational complexity, and prove convergence with spectral accuracy in space and second-order accuracy in time under suitable regularity. The convergence theory is complemented by rigorous semi-discrete and fully discrete error estimates, as well as comprehensive numerical experiments in 1D and 2D that confirm the predicted rates and demonstrate practical efficiency via compressed, block-circulant structures. The results provide a robust, scalable framework for high-precision simulation of quasiperiodic diffusion processes, with potential applications in multiscale and turbulence-related settings where translational symmetry and decay fail to hold.

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.

Convergence analysis of PM-BDF2 method for quasiperiodic parabolic equations

TL;DR

This work tackles the numerical solution of time-dependent quasiperiodic parabolic equations with space-quasiperiodic coefficients by marrying the projection method (PM) in space with the second-order backward differentiation formula (BDF2) in time. The authors develop a PM-BDF2 discretization, analyze its computational complexity, and prove convergence with spectral accuracy in space and second-order accuracy in time under suitable regularity. The convergence theory is complemented by rigorous semi-discrete and fully discrete error estimates, as well as comprehensive numerical experiments in 1D and 2D that confirm the predicted rates and demonstrate practical efficiency via compressed, block-circulant structures. The results provide a robust, scalable framework for high-precision simulation of quasiperiodic diffusion processes, with potential applications in multiscale and turbulence-related settings where translational symmetry and decay fail to hold.

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.
Paper Structure (17 sections, 11 theorems, 127 equations, 1 figure, 4 tables)

This paper contains 17 sections, 11 theorems, 127 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Quasiperiodic function space
  3. Quasiperiodic parabolic equation
  4. Numerical methods
  5. PM and BDF2 scheme
  6. PM-BDF2 discretization
  7. Convergence analysis
  8. Main theorem
  9. Preliminary lemmas
  10. Priori estimate for $U(t)$
  11. Error estimate for the semi-discrete problem
  12. Error estimate for BDF2 scheme
  13. Proof of Theorem \ref{['theory:error_estimate_parabolic']}
  14. Numerical experiments
  15. One-dimensional case
  16. ...and 2 more sections

Key Result

theorem 1

There exists a unique solution of the problem eqn:variational.

Figures (1)

  • Figure 1: Figure \ref{['fig:paraboic:three_modes:space']}: The relationship between the error and the spatial parameter $N$ for solution at $T=0.01$; Figure \ref{['fig:parabolic:three_modes:time']}: The relationship between the error and the time step $\tau$ for solution at $T=0.01$.

Theorems & Definitions (18)

  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • theorem 1
  • theorem 2: Energy estimate
  • proof
  • lemma thmcounterlemma
  • theorem 3
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • lemma thmcounterlemma: Grönwall's inequality
  • ...and 8 more