Local Truncation Error (LTE) for DAE Systems

Overview

Adaptive time-stepping relies on calculating the Local Truncation Error (LTE) to dynamically adjust the step size \(h\). However, traditional textbook LTE formulas were derived for ordinary differential equations (ODEs) of the form \(dx/dt = f(x)\).

Circuit simulators, including PyCircuit, actually solve Differential Algebraic Equations (DAEs) of the form:

\[\frac{d}{dt} q(x(t)) + f(x(t)) + b(t) = 0\]

where \(q(x(t))\) represents nonlinear charge or flux. Applying standard ODE LTE formulas to DAEs is merely an approximation that can lead to accuracy and efficiency losses in stiff, highly non-linear circuits.

DAE-Specific LTE Formulation

Following the derivation in “An Efficient Time Step Control Method in Transient Simulation for DAE System” (Yao et al.), PyCircuit implements the rigorous LTE formulas specific to DAE systems.

By applying Taylor expansion with a Lagrange remainder directly to the charge vector \(q(x(t))\) and substituting it back into the Linear Multi-Step (LMS) equations, we obtain exact analytical formulas for DAE truncation errors.

For example, the DAE LTE for the Trapezoidal Method is given by:

\[\epsilon_T = -\frac{1}{12} (\dot{q}_{x_n} + 0.5 h \ddot{f}_{x_n})^{-1} (g(x_n, t_n) - 2g(x_{n-1}, t_{n-1}) + g(x_{n-2}, t_{n-2})) h\]

This approach fundamentally ensures that PyCircuit calculates the true truncation error of the charge/flux integration, rather than an ODE approximation.

Impact on Simulation

Using the DAE-specific LTE formulation provides two major benefits: 1. Accuracy: Time-step control mathematically matches the true physical dynamics of the non-linear charge nodes. 2. Efficiency: It prevents the solver from taking excessively small steps (which happens when ODE approximations break down during fast nonlinear transitions), ultimately speeding up the simulation without sacrificing precision.