ABSTRACT
Numerical integration is the most common way to solve sets of ordinary differential equations (ODEs), in particular when more than a few ODEs need to be solved simultaneously. This chapter considers the Euler method, begins with a set of initial conditions and then utilizes the instantaneous rate of change at those conditions to estimate the new conditions. The midpoint method uses the Euler method to estimate the variable value and to determine the instantaneous rate of change at that value to generate a slope. The midpoint method is called second order because its error is estimated to be proportional to the cube of the time step. The Runge–Kutta method builds on the Euler and midpoint methods by calculating a weighted average of four slopes. The error associated with the Runge–Kutta method is proportional to the fifth power of the time step, so this method is called fourth order.
