针对多速率偏微分代数方程的线方法的索引分析
Index-analysis for a method of lines discretising multirate partial differential algebraic equations
AI总结：
本文研究多速率偏微分代数方程的线方法，分析其近似DAE系统的微分指数，并通过数值模拟验证结果。
摘要：
在射频应用中，电路生成的信号通常经过幅度调制和/或频率调制。数学建模通常得到微分代数方程（DAEs）系统。多变量信号模型将DAEs转化为多速率偏微分代数方程（MPDAEs）。在频率调制情况下，需要额外条件来确定合适解。我们考虑最优解的必要条件和相位条件。线方法将MPDAEs及额外条件离散化，生成更大的DAE系统。我们分析此近似DAE系统的微分指数，假设原始DAEs为半显式系统。指数取决于额外条件中包含微分变量或代数变量。我们通过示例数值模拟展示结果，并用数值方法验证指数。
英文摘要：
In radio frequency applications, electric circuits generate signals, which are amplitude modulated and/or frequency modulated. A mathematical modelling yields typically systems of differential algebraic equations (DAEs). A multivariate signal model transforms the DAEs into multirate partial differential algebraic equations (MPDAEs). In the case of frequency modulation, an additional condition is required to identify an appropriate solution. We consider a necessary condition for an optimal solution and a phase condition. A method of lines, which discretises the MPDAEs as well as the additional condition, generates a larger system of DAEs. We analyse the differential index of this approximative DAE system, where the original DAEs are assumed to be semi-explicit systems. The index depends on the inclusion of either differential variables or algebraic variables in the additional condition. We present results of numerical simulations for an illustrative example, where the index is also verified by a numerical method.
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