Signal-Flow Based Runge-Kutta Methods for the Simulation of Complex Networks

TL;DR

This paper introduces signal-flow based Runge-Kutta methods that leverage system topology and activity patterns to efficiently simulate complex networks with multirate behavior, reducing computational effort while maintaining accuracy.

Contribution

It proposes novel Runge-Kutta methods utilizing signal flow and system topology to identify inactive regions, enhancing simulation efficiency for complex networks.

Findings

Methods effectively identify inactive regions in networks.

Significant reduction in computation time demonstrated.

Maintains accuracy despite reduced computations.

Abstract

Complex dynamical networks appear in a wide range of physical, biological, and engineering systems. The coupling of subsystems with varying time scales often results in multirate behavior. During the simulation of highly integrated circuits, for example, only a few elements underlie changing signals whereas the major part -- usually up to 80 or even 90 per cent -- remains latent. Standard integration schemes discretize the entire circuit with a single step size which is mainly limited by the accuracy requirements of the rapidly changing subcircuits. It is of a particular interest to speed up the simulation without a significant loss of accuracy. By exploiting the latency of the system, only a fraction of the equations has to be formulated and solved at a given time point. G\"unther and Rentrop suggest that multirate strategies must be based both on the numerical information of the integration scheme and on the topology of the circuit. In this paper, we will introduce a directed graph describing the interdependency of the underlying system and propose Runge--Kutta methods which utilize the signal flow of the system in order to identify and exploit inactive regions. Furthermore, we describe an extension of these methods to identify and exploit periodic subsystems.