Analysis and implementation of TR-BDF2 Original Research Article

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 C1 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.