Index-analysis for a method of lines discretising multirate partial differential algebraic equations

TL;DR

This paper analyzes the differential index of a method of lines discretisation applied to multirate partial differential algebraic equations arising in radio frequency circuit modeling, with numerical verification of the index.

Contribution

It provides a detailed analysis of the differential index for the discretised system and explores how the additional condition affects the index, supported by numerical simulations.

Findings

Index depends on inclusion of differential or algebraic variables in the condition

Discretisation preserves the semi-explicit structure of the original DAEs

Numerical simulations verify the theoretical index analysis

Abstract

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.