Analysis and implementation of TR-BDF2

Bank et al. (1985) developed a one-step method, TR-BDF2, for the simulation of circuits and semiconductor devices based on the trapezoidal rule and the backward differentiation formula of order 2 that provides some of the important advantages of BDF2 without the disadvantages of a memory. Its success and popularity in the context justify its study and further development for general-purpose codes. Here the method is shown to be strongly S-stable. It is shown to be optimal in a class of practical one-step methods, An efficient, globally C-1 interpolation scheme is developed. The truncation error estimate of Bank et al. (1985) is not effective when the problem is very stiff. Coming to an understanding of this leads to a way of correcting the estimate and to a more effective implementation, These developments improve greatly the effectiveness of the method for very stiff problems.