Runge-Kutta MethodsGiven an initial value problem: y ' = f(x,y), y(x0) = y0, 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 + . . . + cs ks
where the ci are constants and k1 = h f(x0,y0), ki = h f(Xi,Yi) where Xi = x0 + ai h and Yi = y0 + bi,1 k1 + . . . + bi,i-1 k1-1, where the ai and bi,j are constants. The constants ci, ai and bi,j are determined by equating like powers of h for the first r powers in the Taylor series expansions of y(x0+h) and y(x0,h). 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 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 th order and (r + 1) st order Runge-Kutta methods in which all of the ai and bi,j constants of the r th order method are ai and bi,j constants of the (r + 1) st order method. The difference between the r th order estimate and the (r + 1) st order estimate yields an estimate of the error. Such pairs of Runge-Kutta methods are called embedded Runge-Kutta methods.
Runge-Kutta methods are stable and convergent methods with region of absolute stability
| 1 + µh + . . . + (1/r!) (µh) r | < 1
in the complex µh plane.