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Stability and Convergence of Strang Splitting Method for the Allen-Cahn Equation with Homogeneous Neumann Boundary Condition

Chaoyu Quan, Zhijun Tan, Yanyao Wu

TL;DR

This work analyzes the Strang splitting method with variable time steps applied to the Allen–Cahn equation under homogeneous Neumann boundary conditions, proposing a framework that yields uniform $H^k$-norm stability and rigorous $H^k$-norm convergence for initial data in $H^{k+6}(\Omega)$. The approach combines maximum-principle arguments, Sobolev interpolation, and exponential-operator techniques to control nonlinear and boundary effects on nonuniform meshes. Numerical experiments using adaptive time stepping verify energy dissipation, $L^{\infty}$-bounds, and near-second-order accuracy, while also demonstrating efficiency gains over uniform stepping. The results extend Strang splitting convergence theory beyond periodic boundaries and provide practical guidance for adaptive schemes in gradient-flow PDEs such as the Allen–Cahn equation.

Abstract

The Strang splitting method has been widely used to solve nonlinear reaction-diffusion equations, with most theoretical convergence analysis assuming periodic boundary conditions. However, such analysis presents additional challenges for the case of homogeneous Neumann boundary condition. In this work the Strang splitting method with variable time steps is investigated for solving the Allen--Cahn equation with homogeneous Neumann boundary conditions. Uniform $H^k$-norm stability is established under the assumption that the initial condition $u^0$ belongs to the Sobolev space $H^k(Ω)$ with integer $k\ge 0$, using the Gagliardo--Nirenberg interpolation inequality and the Sobolev embedding inequality. Furthermore, rigorous convergence analysis is provided in the $H^k$-norm for initial conditions $u^0 \in H^{k+6}(Ω)$, based on the uniform stability. Several numerical experiments are conducted to verify the theoretical results, demonstrating the effectiveness of the proposed method.

Stability and Convergence of Strang Splitting Method for the Allen-Cahn Equation with Homogeneous Neumann Boundary Condition

TL;DR

This work analyzes the Strang splitting method with variable time steps applied to the Allen–Cahn equation under homogeneous Neumann boundary conditions, proposing a framework that yields uniform -norm stability and rigorous -norm convergence for initial data in . The approach combines maximum-principle arguments, Sobolev interpolation, and exponential-operator techniques to control nonlinear and boundary effects on nonuniform meshes. Numerical experiments using adaptive time stepping verify energy dissipation, -bounds, and near-second-order accuracy, while also demonstrating efficiency gains over uniform stepping. The results extend Strang splitting convergence theory beyond periodic boundaries and provide practical guidance for adaptive schemes in gradient-flow PDEs such as the Allen–Cahn equation.

Abstract

The Strang splitting method has been widely used to solve nonlinear reaction-diffusion equations, with most theoretical convergence analysis assuming periodic boundary conditions. However, such analysis presents additional challenges for the case of homogeneous Neumann boundary condition. In this work the Strang splitting method with variable time steps is investigated for solving the Allen--Cahn equation with homogeneous Neumann boundary conditions. Uniform -norm stability is established under the assumption that the initial condition belongs to the Sobolev space with integer , using the Gagliardo--Nirenberg interpolation inequality and the Sobolev embedding inequality. Furthermore, rigorous convergence analysis is provided in the -norm for initial conditions , based on the uniform stability. Several numerical experiments are conducted to verify the theoretical results, demonstrating the effectiveness of the proposed method.

Paper Structure

This paper contains 8 sections, 10 theorems, 133 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Regularity of the exact solution
  3. Stability analysis
  4. Maximum principle
  5. $H^k$-norm stability
  6. Error estimates
  7. Numerical experiments
  8. Conclusions

Key Result

Theorem 2.1

\newlabellemma_derivative0 Assume $u^0 \in H^k(\Omega)$ with $k \ge 0$ and $\|u^0\|_{L^{\infty}}\le 1$. There exists a constant $C\ge 0$ depending on $(\Omega,d,T,\|u^0\|_{H^{k}})$, such that the solution $u(t)$ to eq:governing1 satisfies

Figures (4)

  • Figure 1: Example \ref{['accuracy_test']}: (a) Evolution of energy with final time $T_2=10$. (b) Evolution of maximum norm with final time $T_2=10$. (c) Evolution of time step size with final time $T_2=10$. (d) Evolution of CPU cost time with final time $T_2=10$.
  • Figure 2: Example \ref{['example 3']}: (a) Evolution of energy with final time $T_2=10$. (b) Evolution of maximum norm with final time $T_2=10$. (c) Evolution of CPU cost with final time $T_2=10$. (d) Evolution of time step size with final time $T_2=10$.
  • Figure 3: Example \ref{['example 4']}: Snapshots of $\frac{1}{2}u_1-u_2$ at $t=100.0186,500.0114,1000.0094,2000$ using adaptive time-stepping strategy \ref{['eq:adaptive']}.
  • Figure 4: Example \ref{['example 4']}: (a) Evolution of time step size (in log scale) with final time $T=2000$. (b) Evolution of energy (in log scale) with final time $T=2000$.

Theorems & Definitions (28)

  • Theorem 2.1
  • Proof 1
  • Theorem 3.1
  • Proof 2
  • Theorem 3.2
  • Proof 3
  • Remark 3.3
  • Remark 3.4
  • Lemma 4.1
  • Proof 4
  • ...and 18 more