基于对称思想的时滞次扩散方程若干新数值格式
Some new numerical schemes for delay sub-diffusion equations via the idea of symmetry
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中文总结 AI 辅助
本文针对带时滞的分数阶次扩散方程，提出结合L1与对称分数阶降阶法的新数值格式，实现时间方向$2-\alpha/2$阶最优收敛，并开发二维加权ADI格式及分级网格误差分析框架，兼顾高精度与高效性。
中文摘要 AI 辅助
本文提出了若干新颖的数值格式，这些格式协同地整合了L1方法和对称分数阶降阶（SFOR）方法，专门用于具有时间延迟的分数阶次扩散方程。关键的是，我们严格建立了时间方向上的最优收敛阶为$2 - \alpha/2$，这比在相同光滑性条件下的经典阶$2-\alpha$有所改进。为了进一步提高计算效率，我们针对二维情形开发了一种加权交替方向隐式（ADI）格式，并分析了其与紧致空间离散的耦合，从而在空间上实现了高阶精度。此外，我们引入了一个新的框架来分析在分级时间网格上L1型离散的误差，该框架放宽了网格分级要求，并仍在连续时间点保证最优收敛速率。一些数值实验全面验证了所得格式的理论精度和计算效率。
英文摘要
In this work, we propose several novel numerical schemes that synergistically integrates the $L1$ method and symmetric fractional-order reduction (SFOR) method, specifically designed for fractional sub-diffusion equations with time delay. Crucially, we rigorously establish an optimal convergence order of $2 - α/2$ in temporal direction, which improves upon the classical rate of $2-α$ under identical smoothness conditions. To further enhance computational efficiency, we develop a weighted alternating direction implicit (ADI) scheme for the two-dimensional case and analyze its coupling with compact spatial discretization---thereby achieving high-order accuracy in space. Moreover, we introduce a new framework for analyzing the error of $L1$-type discretization on graded temporal meshes, which relaxes the mesh grading requirement and still guarantees optimal convergence rate at continuous time points. Some numerical experiments comprehensively validate both the theoretical accuracy and computational efficiency of the resulting schemes.
发表机构
- Southern University of Science and Technology（南方科技大学）
- South China Agricultural University（华南农业大学）
- University of Macau（澳门大学）
- Guangdong University of Technology（广东工业大学）
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