<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$H^k$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mi>k</mi> </msup> </math></EquationSource> </InlineEquation>-norm stability is established under the assumption that the initial condition <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$u^0$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>u</mi> <mn>0</mn> </msup> </math></EquationSource> </InlineEquation> belongs to the Sobolev space <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$H^k(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mi>k</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with integer <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$k\ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, using the Gagliardo–Nirenberg interpolation inequality and the Sobolev embedding inequality. Furthermore, rigorous convergence analysis is provided in the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$H^k$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mi>k</mi> </msup> </math></EquationSource> </InlineEquation>-norm for initial conditions <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$u^0 \in H^{k+6}(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>u</mi> <mn>0</mn> </msup> <mo>∈</mo> <msup> <mi>H</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>6</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, based on the uniform stability. Several numerical experiments are conducted to verify the theoretical results, demonstrating the effectiveness of the proposed method.</p>

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

期刊论文
发表日期：

2026年3月24日

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Chaoyu Quan,
Zhijun Tan,
Yanyao Wu

摘要

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 $H^k$ $H^{k}$ -norm stability is established under the assumption that the initial condition $u^0$ $u^{0}$ belongs to the Sobolev space $H^k(\Omega )$ $H^{k} (Ω)$ with integer $k\ge 0$ $k \geq 0$ , using the Gagliardo–Nirenberg interpolation inequality and the Sobolev embedding inequality. Furthermore, rigorous convergence analysis is provided in the $H^k$ $H^{k}$ -norm for initial conditions $u^0 \in H^{k+6}(\Omega )$ $u^{0} \in H^{k + 6} (Ω)$ , based on the uniform stability. Several numerical experiments are conducted to verify the theoretical results, demonstrating the effectiveness of the proposed method.