Advanced Numerical Methods for Multitime Partial Differential–Algebraic Equations in Wireless Circuit Simulation

The simulation of modern wireless communication circuits remains challenging because of the coexistence of nonlinear behavior, heterogeneous subsystems, and widely separated time scales. This review presents a structured overview of advanced numerical methods for solving multitime partial differential–algebraic equations (MPDAEs) arising in circuit-level modeling of RF and microwave systems. Compared with previous survey papers, the main contribution of this work is to organize the literature according to the underlying numerical strategy, distinguishing purely time-domain, hybrid time–frequency, multidimensional frequency-domain, and circuit-block partitioning approaches. The reviewed methods show that multitime formulations can deliver substantial computational gains over conventional simulation techniques, particularly for multirate and multiscale circuits. Time-domain techniques are generally more robust for strongly nonlinear regimes, whereas frequency-domain and hybrid methods are often more efficient when the waveform can be represented with a limited number of harmonics. Circuit-block partitioning further improves efficiency by exploiting active and latent variables, but the computational complexity of MPDAE methods increases rapidly with the number of time scales, and their applicability becomes more limited for aperiodic or highly general multirate excitations. Overall, this review highlights both the strengths and the practical limitations of current MPDAE-based numerical approaches and identifies open challenges for future research.