AI 中文总结
本文提出结合超时间步与龙格-库塔技术的ExtSTS方法，解决多物理场初值问题的时间积分效率问题，其性能优于ARK、Strang分裂等方法。
AI 中文摘要
多物理场初值问题耦合了具有不同稳定性特性的过程，例如平流、扩散和刚性局部反应。标准隐式-显式（ImEx）加性龙格-库塔（ARK）方法可精确处理这些过程，但当扩散与反应归为一组时需要全局耦合的隐式求解；算子分裂可避免此类求解，但通常耦合性较弱且无廉价的时间误差估计；而PIROCK则与特定的龙格-库塔-切比雪夫超时间步（STS）构造绑定。本文提出扩展超时间步（ExtSTS）方法，该系列时间积分方案将扩散项的超时间步方法与其余项的显式、隐式或ImEx龙格-库塔处理相结合。耦合基于多速率无穷小技术，产生可解耦的方法，保留局部隐式求解，支持自适应时间步长的嵌入误差估计，并允许灵活使用现代STS方法。本文介绍ExtSTS方法族，提供创建ExtSTS方法的稳健技术，构建相应的线性稳定性理论，并构造嵌入型ImEx、显式和隐式ExtSTS方法。对一维和二维平流-扩散-反应问题的数值实验表明，ExtSTS方法在各种参数范围和边界条件下均表现稳健，且通常比ARK、Strang分裂和PIROCK方法更高效，尤其在算子间强耦合至关重要时优势显著。
英文摘要
Multiphysics initial-value problems couple processes with distinct stability properties, such as advection, diffusion, and stiff local reactions. Standard implicit-explicit (ImEx) additive Runge--Kutta (ARK) methods can treat these processes accurately, but require globally coupled implicit solves when diffusion is grouped with reaction; operator splitting avoids such solves but typically provides weaker coupling and no inexpensive temporal error estimate; and PIROCK is tied to a specific Runge--Kutta--Chebyshev super-time-stepping (STS) construction. We introduce extended super-time-stepping (ExtSTS) methods, a family of time integration schemes that combine super-time-stepping methods for diffusive terms with explicit, implicit, or ImEx Runge--Kutta treatment of the remaining terms. The coupling is based on multirate infinitesimal techniques, yielding solve-decoupled methods that retain localized implicit solves, support embedded error estimation for adaptive time stepping, and allow flexible use of modern STS methods. We present the ExtSTS method family, provide a robust technique for ExtSTS method creation, formulate the corresponding linear stability theory, and construct embedded ImEx, explicit, and implicit ExtSTS methods. Numerical experiments on one- and two-dimensional advection-diffusion-reaction problems show that ExtSTS methods are robust across parameter regimes and boundary conditions, and are often more efficient than ARK, Strang splitting, and PIROCK methods, especially when strong coupling between operators is important.
