In circuit simulation, we generally deal with stiff problems, i.e., problems with time constants that may vary by multiple orders of magnitude. Implicit time integration methods are employed for this type of problem. In the following, their advantages and drawbacks are briefly discussed.
Backward Euler
As a first-order method, it has the lowest accuracy of all implemented methods. I.e., that a relatively small time step is required to achieve good accuracy. On the other hand, it is a very robust and stable method. In transient cosimulation, the time steps are usually very small and dictated by the 3D mesh. The good stability properties make Backward Euler a good candidate for transient cosimulation where higher-order methods may become instable. On the downside, the low-order approximation has a negative effect on energy conservation. Slight energy dissipation might be observed in the simulation results although, physically, there is no source of dissipation.
Adams
It belongs to the class of linear multistep methods. The second-order method is also called trapezoidal method and is absolutely stable, a property that is not satisfied for higher integration orders of Adams. As a second-order method it is much more accurate than Backward Euler. For very stiff problems, there might be problems with numerical oscillations or ringing. Apart from being non-physical these may also degrade numerical efficiency because very small time steps are required to resolve these numerical oscillations. As a remedy, a slight numerical damping is applied to suppress numerical oscillations. When simulating oscillator circuits, Adams might be better suited than Gear because it tends to be more agile and pick up oscillations more easily.
Gear
The Gear method is also called BDF (Backward Differentiation Formulas) method and belongs to the class of linear multistep methods as Adams. The applied second-order method is absolutely stable, a property that is not satisfied for higher integration orders of Gear. As a second-order method it is much more accurate than Backward Euler and has comparable accuracy to Adams. Unlike Adams, Gear is especially well suited for stiff problems and does not tend to suffer from numerical oscillations. It is therefore used as default method for standard transient simulations.
Note that all three time integration methods are also available as legacy version - an older implementation that applies the same formulas as the new implementation .Currently, the legacy version is selected as default, if the selected time integration method is on "Automatic".
Two different approaches have been implemented, fixed and adaptive time steps.
Fixed
With fixed time steps, constant time steps are applied without any means to control the accuracy of the transient simulation directly. Fast transients such as switching events can only be detected and resolved according to the given time step. On the other hand, if the time step is sufficiently small, good accuracy will usually be achieved. The effort per time step is significantly smaller than for adaptive time steps. To some extent, this compensates the missing flexibility to react on fast or slow transients by time step adaptation.
Adaptive
With adaptive time steps, the accuracy of the transient simulation can be controlled. The local truncation error during transient simulation is estimated via a predictor/corrector step and, based on this error, the time step is adjusted appropriately within limits to achieve the desired transient accuracy. The simulation can react on slow or fast transient efficiently by relaxing or tightening the time step. On the other hand, the numerical effort per time step is significantly higher than for fixed time steps. Furthermore, if there are convergence problems due to rapid transient changes, the simulation can react on this by reducing the time step whereas for fixed time steps the simulation can only be stopped with a convergence error.
Adaptive time steps are applied as default, when the time stepping is on "Automatic". If the simulation with adaptive time steps fails, tolerances and/or time step settings are first relaxed and, this is unsuccessful, fixed time steps are applied.
Before a transient simulation is started, all excitation signals are examined in terms of their spectral contents to establish an overall maximum excitation frequency. It affects the applied range of time steps on the one hand and the upper bound, to which frequency-dependent blocks, represented by S-Parameters, are band-limited on the other hand.
Energy-Based
The spectral density of the excitation signal is calculated via a Fourier transform and the signal is truncated in Frequency Domain such, that the truncated signal accommodates a given portion of the energy of the untruncated signal, the so-called energy threshold. This estimator works well for relatively smooth signals. However, if there a very sharp rise- or fall times and quite long hold times, the transitions might be not well resolved and the maximum excitation frequency tends to be underestimated.
Transitiontime-Based
This estimator is especially well suited for applications dealing with digital pulses. The maximum excitation frequency is calculated by putting the amplitude levels of the signal (low/high) in relation to the transition times between different amplitude levels. Of course, the evaluation is difficult, if pulses are defined with infinitely steep flanks.
As default, if the maximum frequency estimator is on "Automatic", the transitiontime-based estimator is applied, if digital pulse signals are present as excitation. Otherwise the energy-based estimator is applied. Furthermore, there is a further option to set the maximum excitation frequency manually.
See also