eldo_user

Speed and Accuracy
Integration Methods
Integration Methods
The process of ‘eliminating’ the time-derivatives in the original circuit equations is a source of
error.
Tip
See “Speed and Accuracy in Eldo” on page 1245.
These time-derivatives are replaced with finite differences, which only approximate the true
time-derivatives. One of the most basic approximation schemes is the Backward-Euler method,
which approximates v’(ti) using the finite difference (v(ti)-v(ti-1)) / (ti-ti-1). Intuitively, it is
easy to understand that the smaller the time step (h=ti-ti-1) the smaller the error. More
sophisticated schemes exist, which provide less error with the same time step, such as the socalled
trapezoidal and Gear methods.
The numerical resolution of differential equations is a vast subject. Dozens of methods have
been proposed and studied in depth, some of them having very attractive properties, particularly
allowing the use of very large time steps without encountering stability issues. However, in the
context of circuit simulation, many of these methods are not practically applicable. The
differential equations that model an electrical network (either IC or PCB) dynamics
unfortunately have undesirable characteristics. For example, they are implicit (in most if not all
formulations used in commercial simulators) and very often ‘stiff’ (which means that the
individual time constants involved in the system can routinely exhibit orders of magnitude
differences). The cost of evaluation of the non-linear functions is also very high. This
unfortunate situation leaves very few good practical candidates as integration methods.
Eldo implements three integration methods, namely the simple Backward-Euler (BE) method,
the trapezoidal method (TRAP), and the Gear method (GEAR). The trapezoidal method is the
default in Eldo; it provides a very good speed/accuracy compromise. TRAP and GEAR are both
second-order methods, whereas BE is a first-order method. The accuracy of BE is theoretically
one order of magnitude worse than TRAP or GEAR, so it is usually reserved for cases where
speed is the most important criterion, and accuracy, to a certain extent, can be sacrificed. In
many cases, the speed increase obtained with BE is not considerable, although the loss in
accuracy can be significant. This is because the GEAR and TRAP methods are still relatively
simple methods (the derivative approximations are not too complicated). Thus the accuracy
improvement with TRAP or GEAR over BE is generally worth the increased CPU time.
Although TRAP and GEAR have similar theoretical accuracy, they still have their own
characteristics. TRAP has the very undesirable tendency to generate numerical ‘ringing’,
particularly on the current variables. Ringing shows up as obviously non-physical oscillations
of small (or large!) amplitude riding on top of a correct and accurate average value. The
oscillations have the rhythm of the time steps (they don’t belong to the circuit intrinsic time
constant(s)) and they usually dampen over time if the simulator properly controls the time step.
This does not happen systematically with all test cases, but it is a well-known artifact of the
TRAP method. All simulators, including Eldo, implement some code that tries to eliminate or
reduce these oscillations, but sometimes it may be very difficult to eliminate them entirely.
1246
Eldo® User's Manual, 15.3