Numerical solution of ordinary differential equations

Ordinary differential equations are ubiquitous in science and engineering: in geometry and mechanics from the first examples onwards (Newton, Leibniz, Euler, Lagrange), in chemical reaction kinetics, molecular dynamics, electronic circuits, population dynamics, and many more application areas. They also arise, after semidiscretization in space, in the numerical treatment of time-dependent partial differential equations, which are even more impressively omnipresent in our technologically developed and financially controlled world. The standard initial value problem is to determine a vector-valued function y : [t0, T ] → R with a given initial value y(t0) = y0 ∈ R such that the derivative y′(t) depends on the current solution value y(t) at every t ∈ [t0, T ] in a prescribed way: