A widely used class of methods for the numerical solution of ordinary differential equations. For the initial-value problem y′ = f(x,y), y(x0) = y0, the general form of the m-stage method is i = 1,2,…,mxn + 1 = xn + h The derivation of suitable parameters aij, bi, and ci requires extremely lengthy algebraic manipulations, except for small values of m.
y′ = f(x,y), y(x0) = y0,
i = 1,2,…,m
xn + 1 = xn + h
Some early examples were developed by Runge and a systematic treatment was initiated by Kutta about 1900. Recently, significant advances have been made in the development of a general theory and in the derivation and implementation of efficient methods incorporating error estimation and control.
Except for stiff equations (see ordinary differential equations), explicit methods with aij = 0, j ≥ i are used. These are relatively easy to program and are efficient compared with other methods unless evaluations of f(x,y) are expensive.
with aij = 0, j ≥ i
To be useful for practical problems, the methods should be implemented in a form that allows the stepsize h to vary across the range of integration. Methods for choosing the steps h are based on estimates of the local error. A Runge-Kutta formula should also be derived with a local interpolant that can be used to produce accurate approximations for all values of x, not just at the grid-points xn. This avoids the considerable extra cost caused by artificially restricting the stepsize when dense output is required.