In numerical analysis, the Runge–Kutta methods are an important family of implicit and explicit iterative methods for the approximation of solutions of ordinary differential equations.
The common fourth-order Runge–Kutta method
One member of the family of Runge–Kutta methods is so commonly used that it is often referred to as “RK4” or simply as “the Runge–Kutta method”. Let an initial value problem be specified as follows.
, with .
We equally split the interval into subintervals by the following interior points
Of course, . The number , which will be denoted by , is called the step size. The value of at step is usually denoted by , that is, . Therefore, the initial value problem can be rewritten as , where and with . The Runge–Kutta 4 method tells us that, having the value of at , we can compute the value of at by using the following scheme
Thus, for each step, we need to compute all four , , constants.
The Runge–Kutta 4 method for a system of ODEs
Now we investigate the question in the title of this entry. An th-order system of 1st initial value problems has the form
for , with the initial conditions
Let be . We denote by the following number
At each step , having all we can compute for all . To do this, for each , we need to compute all , . More precisely, we need to compute
As can be seen, at step and for each , all the values , with must be computed before any of the expressions .
We now consider the following example
Therefore, at the step , we can compute and by using the following scheme