Time Discretization vs. State Quantification: Activity Homogeneity and Discontinuities

Abstract

In this work, we define the concept of activity homogeneity for the solutions of Ordinary Differential Equations (ODEs). This indicator quantifies the similarity in the rate of change of the different variables in the system over time. We also show that this measure provides useful criteria for determining whether it is more convenient to use classic numerical integration methods based on time discretization or state quantification based methods. In addition, we extend the analysis to discontinuous systems and the effects of the presence of events in each type of numerical integration scheme. Finally, we apply the developed concepts to two case studies: an advection - diffusion - reaction system (corresponding to a continuous model) and a neural network (corresponding to a hybrid model). We compare the theoretical results with those obtained from simulations of both systems using different numerical integration methods.