Co-simulation errors due to step size changes

TL;DR

The paper reveals that nonuniform step-size changes in co-simulation can introduce additional coupling errors, even when overall steps shrink. Through analytic treatment of two-subsystem models, it derives leading-order expressions showing how step-size differences and interfacial flows drive a persistent state discrepancy. Two illustrative examples—the damped harmonic oscillator and connected fluid reservoirs—demonstrate the phenomenon and its sensitivity to how step sizes are changed over time. The work offers practical mitigation strategies and highlights limitations, pointing to future work on higher-order extrapolation and implicit co-simulation methods. Overall, it provides a nuanced view of step-size effects in co-simulation beyond simple step-size magnitude considerations.

Abstract

When two simulation units in a continuous-time co-simulation are connected via some variable $q$, and both simulation units have an internal state which represents the time integral of $q$, there will generally be a discrepancy between those states due to extrapolation errors. Normally, such extrapolation errors diminish if the macro time step size is reduced. Here we show that, under certain circumstances, step size changes can cause such discrepancies to increase even when the change is towards smaller steps.

Figures (10)

• Figure 1: Solution of the damped harmonic oscillator with $m = c = k = 1$ and initial conditions $x(0) = 1$ and $\dot x(0) = 0$.
• Figure 2: Co-simulation setup of the damped harmonic oscillator.
• Figure 3: Co-simulated solutions of the harmonic oscillator. The top panels show the solution obtained with a fixed step size; the lower are for the variable-step-size algorithm. The leftmost panels show the entire simulation; the rightmost plots show a zoomed-in picture of the tail ($9 \le t \le 15$).
• Figure 4: The upper panel shows the discrepancy between the displacement states in $S_1$ and $S_2$, $\Delta x = x_1 - x_2$, in the fixed-step-size and variable-step-size cases. The lower panel shows the macro time step size. The simulation was run with an initial velocity $\dot x_1 = \dot x_2 = 0$.
• Figure 5: The upper panel shows the discrepancy between the displacement states in $S_1$ and $S_2$, $\Delta x = x_1 - x_2$, in the fixed-step-size and variable-step-size cases. The lower panel shows the macro step size. Unlike in \ref{['fig:harmonic-oscillator-discrepancy']}, the simulation was run with an initial velocity $\dot x_1 = \dot x_2 = 1$.