Index-analysis for a method of lines discretising multirate partial differential algebraic equations
For researchers working on numerical simulation of RF circuits modeled by DAEs, this work provides an index analysis for a discretization approach, but it is incremental as it extends known index concepts to a specific class of MPDAEs.
The paper analyzes the differential index of a method of lines discretization for multirate partial differential algebraic equations (MPDAEs) arising from modulated signals in RF circuits, showing that the index depends on whether differential or algebraic variables are included in the additional phase condition. Numerical simulations verify the index analysis.
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.