The methods of Runge-Kutta are methods of numerical Analyze of approximation of solutions of differential equations. They bear the name of the mathematicians Carl Runge and Martin Wilhelm Kutta. These methods rest on the principle of the Itération, i.e. a first estimate of the solution is used to calculate one second estimate, more precise, and so on.

Method of Runge-Kutta of order 1 (RK1)

This method is equivalent to the Méthode of Euler.

Let us consider the following problem:

$y\text{'}\; =\; F\; (T,\; there),\; \backslash \; quad\; there\; (t\_0)\; =\; y\_0$

Method RK1 is given by the equation

$y\_\; \{n+1\}\; =\; y\_n\; +\; H\; F\; \backslash \; left\; (X,\; y\_n\; \backslash \; right)$

where $h$ is the step of the iteration.

Method of Runge-Kutta traditional of order four (RK4)

It is a particular case of very frequent use, indicated RK4.

Let us consider the following problem:

$y\text{'}\; =\; F\; (T,\; there),\; \backslash \; quad\; there\; (t\_0)\; =\; y\_0$

Method RK4 is given by the equation:

$y\_\; \{n+1\}\; =\; y\_n\; +\; \{H\; \backslash \; over\; 6\}\; (k\_1\; +\; 2k\_2\; +\; 2k\_3\; +\; k\_4)$

where

$k\_1\; =\; F\; \backslash \; left\; (t\_n,\; y\_n\; \backslash \; right)$

$k\_2\; =\; F\; \backslash \; left\; (t\_n\; +\; \{H\; \backslash \; over\; 2\},\; y\_n\; +\; \{H\; \backslash \; over\; 2\}\; k\_1\; \backslash \; right)$

$k\_3\; =\; F\; \backslash \; left\; (t\_n\; +\; \{H\; \backslash \; over\; 2\},\; y\_n\; +\; \{H\; \backslash \; over\; 2\}\; k\_2\; \backslash \; right)$

$k\_4\; =\; F\; \backslash \; left\; (t\_n\; +\; H,\; y\_n\; +\; H\; k\_3\; \backslash \; right)$

The idea is that the following value ( there _{N +1}) is approached by the sum of the current value ( there _N ) and of the product of the size of the interval ( H ) by the estimated slope. The slope is obtained by a weighted average of slopes:

• K ₁ is the slope at the beginning of the interval;

• K ₂ is the slope in the middle of the interval, by using the slope K ₁ to calculate the value of there at the point T _N + H /2 by the means of the Méthode of Euler;
• K ₃ is again the slope in the middle of the interval, but obtained this time by using the slope K ₂ to calculate there ;
• K ₄ is the slope at the end of the interval, with the value of there calculated by using K ₃.

In the average of the four slopes, a larger weight is given to the slopes at the point medium.

$\backslash \; mbox\; \{slope\}\; =\; \backslash \; frac\; \{k\_1\; +\; 2k\_2\; +\; 2k\_3\; +\; k\_4\}\; \{6\}.$

Method RK4 is a method of order 4, which means that the error made with each stage is about H ⁵, whereas the accumulated total error is about H ⁴.

These formulas are also valid for functions with vectorial values.

See too

• Method of Newmark