AI 中文总结
针对系统频谱随时间显著变化的多尺度建模问题，提出广义投影积分（GPI）格式，经实验验证其在多维度性能上优于现有方法。
AI 中文摘要
当系统的频谱随时间显著变化时，宏观、介观和微观时间步长以及爆发长度的固定选择会变得不足，需要自适应且局部知情的策略。为应对这些挑战，本文提出一种新颖且灵活的广义投影积分（GPI）格式，旨在适应随时间变化的频谱变化和动态演化的尺度分离。该框架统一并扩展了多种现有多尺度方法，从而提供更通用和适应性更强的计算范式。对GPI格式进行了全面的稳定性分析，包括对稳定区域分裂的详细研究，这是本工作的核心组成部分。此外，开发了依赖于问题的微观、介观和宏观时间步长以及爆发长度的选择策略，并通过数值实验系统验证了其影响。为评估所提格式的有效性，考虑了三个具有不同类型频谱演化的代表性问题，基于（i）微观时间步长数量、（ii）精度、（iii）计算时间、（iv）内存使用和（v）微观尺度模拟的比例，将所提GPI格式的性能与多种现有投影积分方法以及一些广泛使用的刚性求解器进行了评估和比较。结果表明，GPI格式在性能上始终优于现有方法。
英文摘要
When the spectrum of a system varies significantly over time, fixed choices of macro-, meso-, and micro-time steps, as well as burst lengths, become inadequate, necessitating adaptive and locally informed strategies. To address these challenges, this article proposes a novel and flexible generalised projective integration (GPI) scheme, designed to accommodate time-dependent spectral variation and dynamically evolving scale separation. The proposed framework unifies and extends several existing multiscale methodologies, thereby offering a more general and adaptable computational paradigm. A comprehensive stability analysis of the GPI scheme is carried out, including a detailed investigation of the splitting of the stability region, which forms a central component of this work. Furthermore, problem-dependent strategies for selecting the micro-, meso-, and macro-time steps, as well as the burst length, are developed and their impacts are systematically validated through numerical experiments. To assess the effectiveness of the proposed scheme, three representative problems with distinct types of spectral evolution are considered. The performance of the proposed GPI scheme is evaluated and compared with several existing projective integration methods as well as some widely used stiff solvers, based on (i) number of micro time steps, (ii) accuracy, (iii) computational time, (iv) memory usage and (v) proportion of micro-scale simulations. The results demonstrate that the GPI scheme consistently outperforms the existing methods in terms of performance.
Comments47 pages, 12 figures, 15 tables
