The Euler method is a first-order numerical procedure for solving ordinary differential equations
(ODEs) with a given initial value. It is the most basic explicit method for numerical integration
of ordinary differential equations and is the simplest Runge-Kutta method. The Euler method is a
first-order method, which means that the local error (error per step) is proportional to the
square of the step size, and the global error (error at a given time) is proportional to the step
size. It also suffers from stability problems. For these reasons, the Euler method is not often
used in practice. It serves as the basis to construct more complicated methods.
D. Greenspan, Numerical Solution of Ordinary Differential Equations: for Classical,
Relativistic and Nano Systems, 1st ed, Wiley-VCH, 2006.