标题： 奇异分裂方法在具有齐次诺伊曼边界条件的阿伦-卡恩方程中的稳定性和收敛性 显示英文标题

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

摘要： 斯特朗分裂方法已被广泛用于求解非线性反应扩散方程，大多数理论收敛分析都假设周期性边界条件。 然而，对于齐次诺伊曼边界条件的情况，这种分析带来了额外的挑战。 在本工作中，研究了具有可变时间步长的斯特朗分裂方法，以求解具有齐次诺伊曼边界条件的阿伦-卡恩方程。 在假设初始条件$u^0$属于 Sobolev 空间$H^k(\Omega)$，其中整数$k\ge 0$的情况下，利用 Gagliardo--Nirenberg 插值不等式和 Sobolev 嵌入不等式建立了均匀$H^k$-范数稳定性。 此外，基于均匀稳定性，对初始条件$u^0 \in H^{k+6}(\Omega)$在$H^k$-范数下提供了严格的收敛分析。 进行了若干数值实验以验证理论结果，证明了所提出方法的有效性。 显示英文摘要

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 $H^k$-norm stability is established under the assumption that the initial condition $u^0$ belongs to the Sobolev space $H^k(\Omega)$ 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 $u^0 \in H^{k+6}(\Omega)$, based on the uniform stability. Several numerical experiments are conducted to verify the theoretical results, demonstrating the effectiveness of the proposed method.