Discrete Events

ODEs describe the continuous evolution of a system over time, but many systems also involve discrete events that occur at specific times. For example, in a compartmental model of drug delivery, the administration of a drug is a discrete event that occurs at a specific time. In a bouncing ball model, the collision of the ball with the ground is a discrete event that changes the state of the system. It is normally difficult to model these events using ODEs alone, as they require a different approach to handle the discontinuities in the system. While we can represent discrete events mathematically using delta functions, many ODE solvers are not designed to handle these discontinuities, and may produce inaccurate results or fail to converge during the integration.

DiffSol provides a way to model discrete events in a system of ODEs by maintaining an internal state of each solver that can be updated when a discrete event occurs. Each solver has an internal state that holds information such as the current time \(t\), the current state of the system \(\mathbf{y}\), and other solver-specific information. When a discrete event occurs, the user can update the internal state of the solver to reflect the change in the system, and then continue the integration of the ODEs as normal.

DiffSol also provides a way to stop the integration of the ODEs, either at a specific time or when a specific condition is met. This can be useful for modelling systems with discrete events, as it allows the user to control the integration of the ODEs and to handle the events in a flexible way.

The Solving the Problem and Root Finding sections provides an introduction to the API for solving ODEs and detecting events with DiffSol. In the next two sections, we will look at two examples of systems with discrete events: compartmental models of drug delivery and bouncing ball models, and solve them using DiffSol using this API.