AI 中文总结
针对非时间可逆动力学分裂方法的阶数障碍，提出广义改进分裂方法，通过多指标级数形式推导阶数条件并构造6阶方法，提供自动化脚本并关联相关级数形式。
AI 中文摘要
我们研究涉及无界算子的偏微分方程的分裂方法。对于耗散系统等非时间可逆动力学，负分裂系数通常不可取，因为它们需要时间反向步进，当要求所有系数为正时会导致阶数障碍。我们引入广义改进分裂方法以克服该障碍。为分析其收敛性，我们开发了相应的多指标级数形式，为推导阶数条件提供系统框架。利用该形式，我们推导了阶数条件并构造了一个6阶广义改进分裂方法。我们还提供了Python脚本，可自动生成和验证阶数条件，以及构建新的广义改进分裂方法。最后，我们建立了所引入的多指标级数与文献中相关级数形式的联系，特别是词级数、李-巴特彻级数和多指标巴特彻级数。
英文摘要
We consider splitting methods for partial differential equations involving unbounded operators. For non-time-reversible dynamics, such as dissipative systems, negative splitting coefficients are generally not admissible because they require stepping backward in time, leading to an order barrier when all coefficients are required to be positive. We introduce generalized modified splitting methods to overcome this barrier. To analyze their convergence, we develop the corresponding multi-index series formalism, which provides a systematic framework for deriving order conditions. Using this formalism, we derive the order conditions and construct a generalized modified splitting method of order $6$. We also provide Python scripts that automate the generation and verification of order conditions, as well as the construction of new generalized modified splitting methods. Finally, we establish connections between the introduced multi-index series and related series formalisms from the literature, in particular, word series, Lie-Butcher series, and multi-index Butcher series.
Comments30 pages
