the general form of the m-stage method is i = 1,2,…,m xn+1 = xn + h
The derivation of suitable parameters aij, bi, and ci requires extremely lengthy algebraic manipulations, except for small values of m.
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.
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.
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