# Convergence of explicitly coupled Simulation Tools (Cosimulations)

## TL;DR

This paper proves the convergence of explicit cosimulation schemes used in engineering, discusses balance errors due to subsystem coupling, and analyzes the stability issues of balance correction methods.

## Contribution

It provides a convergence proof for explicit cosimulation schemes and analyzes their stability, including the effects of balance correction techniques.

## Key findings

- Convergence is proven for explicit cosimulation schemes.
- Balance errors can accumulate and affect accuracy.
- Balance correction methods can mitigate but not eliminate errors.

## Abstract

In engineering, it is a common desire to couple existing simulation tools together into one big system by passing information from subsystems as parameters into the subsystems under influence. As executed at fixed time points, this data exchange gives the global method a strong explicit component. Globally, such an explicit cosimulation schemes exchange time step can be seen as a step of an one-step method which is explicit in some solution components. Exploiting this structure, we give a convergence proof for such schemes. As flows of conserved quantities are passed across subsystem boundaries, it is not ensured that systemwide balances are fulfilled: the system is not solved as one single equation system. These balance errors can accumulate and make simulation results inaccurate. Use of higher-order extrapolation in exchanged data can reduce this problem but cannot solve it. The remaining balance error has been handled in past work by recontributing it to the input signal in next coupling time step, a technique labeled balance correction methods. Convergence for that method is proven. Further, a proof for the lack of stability of such methods is given for cosimulation schemes with and without balance correction.