Discrete-variable methods
In discrete-variable methods, we are concerned with finding approximations to the solutions, but only at a discrete set of points in the domain. These points could be predetermined before solving, or we could generate them dynamically as part of the integration, to better suit the properties of the functions involved. This is especially useful when the solutions present singularities, for example, once a discrete set of points have been generated, we can compute a nice analytic approximation to the solutions, by a simple process of interpolation.
We have two schema for discrete-variable methods:
One-step methods: The value of the solution at one point is computed solely from information on the previous point. Classical exponents of this scheme are, for instance, Euler's method, improved Euler's method, any second-order two-stage method, or any Runge-Kutta method.
Multistep methods: The value of the solution at one point depends on the knowledge of several previous...
