新型高效隐式-显式延迟修正方法
New Efficient Implicit-Explicit Deferred Correction methods
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中文总结 AI 辅助
该研究提出一种高效IMEX DeC改进方案，通过迭代插值匹配精度与离散化阶数，兼具计算优势与p阶自适应特性，经数值实验验证可高效求解刚性ODE与高阶PDE
中文摘要 AI 辅助
在本研究中，我们探究用于近似常微分方程（ODE）的隐式-显式（IMEX）任意高阶延迟修正（DeC）方法。这类格式的特点是采用迭代过程，每次迭代将精度阶数提高一阶。更确切地说，我们研究一种高效改进方案，该方案通过在连续迭代间引入插值过程，使每次迭代达到的精度与所用离散化阶数系统匹配。一方面，该改进方案带来计算优势，因为低阶迭代可在成本更低的低阶离散化结构上执行；另一方面，它赋予方法天然的p阶自适应特性，这在实际应用场景中极具吸引力。我们针对两类DeC格式研究该改进方案，给出数值验证、效率评估及稳定区域图。数值验证包含多个涉及刚性ODE和带有高阶空间导数的偏微分方程（PDE）的算例，数值实验证实了改进格式以降低的计算成本提供高保真结果的能力，以及自适应策略的有效性。
英文摘要
In this work, we investigate implicit-explicit (IMEX) arbitrary high-order Deferred Correction (DeC) methods for the approximation of ordinary differential equations (ODEs). Such schemes are characterized by an iterative procedure that increases the order of accuracy by one at each iteration. More precisely, we study an efficient modification based on the introduction of interpolation processes between consecutive iterations, with the aim of systematically matching the accuracy achieved at each iteration with the order of the discretization employed. On the one hand, this modification leads to computational advantages, since the low-order iterations are performed on cheaper lower-order discretization structures; on the other hand, it endows the methods with a natural $p$-adaptive character, which is particularly appealing in the context of practical applications. We investigate this modification for two families of DeC schemes, providing numerical validation, efficiency assessments, and stability region plots. The numerical validation includes several examples involving stiff ODEs and partial differential equations (PDEs) with high-order spatial derivatives. The ability of the modified schemes to provide high-fidelity results at reduced computational cost, as well as the effectiveness of the adaptive strategy, is demonstrated through the numerical experiments.
