## Embedded Runge-Kutta Methods

A Runge-Kutta method is a one-step method for approximating the solution*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.

_{i,j}An embedded Runge-Kutta method is a method in which two Runge-Kutta estimates are obtained using the same auxiliary functions

*k*but with a different linear combination of these functions so that one estimate has an order one greater than the other. This allows an error estimate: Suppose that Y is one estimate of order r and Z is the other estimate of order

_{i}r + 1, then the difference between the two estimates satifies

*Z - Y = A h*.

^{ r}+ O(h^{ r + 1})If the step size is scaled by

*µ*so that the new step size is

*(µ h)*then

*Z - Y*becomes

*Z - Y = A (µ h)*.

^{ r}+ O(h^{ r + 1})If

*ε*is the allowable error on the interval from

*(x0, x0 + h)*,

*µ*can be chosen so that the final error is less than that

*ε*.