$A$-Stable and $L$-Stable High-Order Integration Methods for Solving Stiff Differential Equations

摘要

This paper describes a new $A$- and $L$-stable integration method for simulating the time-domain transient response of nonlinear circuits. The proposed method, which is based on the Obreshkov formula, can be made of arbitrary high order while maintaining the $A$-stability property. The new method allows for the adoption of higher order integration methods for the transient analysis of electronic circuits while enabling them to take larger step sizes without violating the stability, leading to faster simulations. The method can be run in an $L$-stable mode to handle circuits with extremely stiff equations. Necessary theoretical foundations, implementation details, error-control mechanisms, and computational results are presented.