线性微分代数方程的围道积分方法与结构摄动
Contour integral methods and structured perturbations for linear differential-algebraic equations
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中文总结 AI 辅助
研究线性微分代数方程时间积分，推广围道积分方法，通过拉普拉斯变换及求积规则重构时域解，还针对线性参数DAE提出调整积分围道策略，经实验验证该方法高效准确，利于模型降阶。
中文摘要 AI 辅助
我们将围道积分方法（CIM）框架推广到线性动力系统的时间积分，该系统在演化过程中始终受代数约束。所提方法通过对与线性微分代数方程（DAE）相关的柯西问题应用拉普拉斯变换，随后通过合适的求积规则近似逆拉普拉斯变换来重构时域解。这为经典龙格 - 库塔方法提供了高效准确的替代方案，经典方法应用于DAE时精度会降低。在论文第二部分，我们处理线性参数DAE，提出一种在CIM框架中使用合适的结构 - 非结构伪谱计算来调整积分围道的有效策略。这能识别出一个能近似整个参数解族的单一积分轮廓，便于高效应用模型降阶技术。最后，通过数值实验验证所提方法并支持理论结果。
英文摘要
We generalize the contour integral methods (CIM) framework to the time integration of linear dynamical systems that are subject to algebraic constraints at all times during their evolution. The proposed approach relies on applying the Laplace transform to the Cauchy problem associated with a linear system of differential-algebraic equations (DAE), and subsequently reconstructing the time-domain solution by approximating the inverse Laplace transform via a suitable quadrature rule. This procedure yields an efficient and accurate alternative to classical Runge-Kutta schemes, which are well known to exhibit order reduction in accuracy when applied to DAE. In the second part of the paper, we address linear parametric DAE and propose an efficient strategy for tuning the integration contour in the CIM framework using suitable structured-unstructured pseudospectral computations. This allows the identification of a single integration profile capable of approximating an entire family of parametric solutions, thereby facilitating the efficient application of model order reduction techniques. Finally, numerical experiments are presented to validate the proposed methodology and support the theoretical findings.
