Rigorous and non-rigorous error estimates for numerical solutions of initial value problems

Two types of error estimation are considered. First some some heuristic error estimates are discussed, such as the explicit Euler method and general Runge-Kutta methods, as well as Richardson extrapolation. Second a reconstruction method using Euler's method and the Crank-Nicolson method is disscussed as a means of a rigorous error bound. The end result is an guaranteed error bound for an intial value problem.