On exponential fitting for circuit simulation

The stability and accuracy properties of exponentially fit integration algorithms applied to the test problem x˙=-Ax are compared with the more standard backward-Euler and semi-implicit methods. For the analysis, A∈IR^n×n is assumed to be connectedly diagonally dominant with positive diagonals, as this models the equations resulting from the way MOS transistors and interconnect parasitics are treated in circuit-level timing simulation programs. Examples are used to demonstrate that all the exponential-fitting methods, and the semi-implicit methods, are much less accurate than backward-Euler for tightly coupled stiff problems, and an example is given which destabilizes one of the exponential-fitting methods. It is then proved that in the limit of large time steps, the more stable exponential-fitting methods become equivalent to a semi-implicit algorithm. It is shown that the backward-Euler, semi-implicit, and certain exponentially fit algorithms are multirate A-stable