What are discretized equations?

Discretized equations are derived from continuous models through the process of discretization, which involves approximating a continuous function or equation by breaking it down into finite parts. This process is essential in numerical analysis and computational methods where continuous variables are sampled at discrete intervals.

In many engineering and scientific contexts, continuous systems (like fluid flow, thermal conduction, etc.) are modeled mathematically with differential equations. To solve these equations using computational tools, adjustments are made to simplify complex mathematical relationships into a form that can be tackled using numerical methods. These often involve replacing continuous derivatives with finite differences, thereby allowing calculations to be performed at specific points or grid locations, resulting in a set of discrete equations.

This approach is fundamental in simulations, modeling, and various applications in engineering analysis where continuous change needs to be approximated via discrete steps or values.