Embedded Runge-Kutta Methods

Description

A Runge-Kutta method is a one-step method for approximating the solution y(x₀+h) of the initial value problem with the form
y(x₀,h) = y₀ + c₁ k₁ + . . . + c_s k_swhere the c_i are constants and k₁ = h f(x₀,y₀), k_i = h f(X_i,Y_i) where X_i = x₀ + a_i h and Y_i = y₀ + b_i,1 k₁ + . . . + b_i,i-1 k_1-1, where the a_i and b_i,j are constants.

An embedded Runge-Kutta method is a method in which two Runge-Kutta estimates are obtained using the same auxiliary functions k_i 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 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 becomesZ - Y = A (µ h)^r + O(h^{r + 1}). If e is the allowable error on the interval from (x0, x0 + h), µ can be chosen so that the final error is less than that e.