## Runge-Kutta Methods

Given an initial value problem:*y ' = f(x,y), y(x*, a Runge-Kutta method is a one-step method for approximating the solution

_{0}) = y_{0}*y(x*of the initial value problem with the form

_{0}+h)*y(x*

_{0},h) = y_{0}+ c_{1}k_{1}+ . . . + c_{s}k_{s}where the

*c*are constants and

_{i}*k*,

_{1}= h f(x_{0},y_{0})*k*where

_{i}= h f(X_{i},Y_{i})*X*and

_{i}= x_{0}+ a_{i}h*Y*, where the

_{i}= y_{0}+ b_{i,1}k_{1}+ . . . + b_{i,i-1}k_{1-1}*a*and

_{i}*b*are constants. The constants

_{i,j}*c*,

_{i}*a*and

_{i}*b*are determined by equating like powers of

_{i,j}*h*for the first

*r*powers in the Taylor series expansions of

*y(x*and

_{0}+h)*y(x*. The result is underdetermined and one is free to set some of the constants and then determine the remaning. The order of the method is

_{0},h)*r*. For

*r < 5*, the constants can be chosen so that

*s = r*, for

*r = 5,6*, the constants can be chosen so that

*s = 6,7*and for

*r > 6*, the constants can be chosen so that

*s = r + 2*. The number

*s*is the number of stages.

It is possible create an

*r*order and

^{ th}*(r + 1)*order Runge-Kutta methods in which all of the

^{ st}*a*and

_{i}*b*constants of the

_{i,j}*r*order method are

^{ th}*a*and

_{i}*b*constants of the

_{i,j}*(r + 1)*order method. The difference between the

^{ st}*r*order estimate and the

^{ th}*(r + 1)*order estimate yields an estimate of the error. Such pairs of Runge-Kutta methods are called embedded Runge-Kutta methods.

^{ st}Runge-Kutta methods are stable and convergent methods with region of absolute stability

| 1 + µh + . . . + (1/r!) (µh)

^{ r}| < 1in the complex µh plane.