Runge and his legacy

Abstract

Carl David Tolmé Runge was a German mathematician. In 1895 he brought about a revolution in the science of what has become numerical analysis. Numerical approximation to the solution of ordinary differential equations had been based on the method of Euler. but it suffered from a low convergence rate. Runge’s discovery in his 1895 paper brought the so-called order from 1 up to 2. This development was followed within a few years by extensions of his techniques which brought the possible order up to 4. The analysis of order conditions for early work on Runge–Kutta methods was based on the use of a scalar test problem y′ = f (x, y), but for high order method derivations and other applications, it is appropriate to use instead an autonomous high dimensional test problem y′ = f (y).Because of the special requirements of stiff problems, implicit Runge–Kutta methods have an importance of their own and these developments are briefly surveyed. The intricate manipulations required to analyse the order of complicated methods has lead to an algebraic theory and its formulation as B-series. Finally, the special applications to structure-preserving algorithms are introduced.