In numerical analysis, the Runge-Kutta methods are a family of techniques for the approximation of solutions of ordinary differential equations. These techniques were developed around 1900 by the mathematicians C. Runge and M.W. Kutta. The fourth-order formulation ("RK4") is the most commonly used, since it provides substantial accuracy without excessive complexity.

Let an initial value problem be specified as follows.

The RK4 method for this problem is given by the following equation:

where

Thus, the next value (y_n+1) is determined by the present value (y_n) plus the product of the size of the interval (h) and an estimated slope. The slope is a weighted average of slopes:

• k₁ is the slope at the beginning of the interval;
• k₂ is the slope at the midpoint of the interval, using slope k₁ to determine the value of y at the point t_n + h/2 using Euler's method;
• k₃ is again the slope at the midpoint, but now using the slope k₂ to determine the y-value;
• k₄ is the slope at the end of the interval, with its y-value determined using k₃.

When the four slopes are averaged, more weight is given to the slopes at the midpoint:

Iterative methods in general may be represented by the generic form y_n+1 = c y_n, where c is a coefficient that depends upon the method used and the equation being evaluated. The primary reason that the RK4 method is successful is that the coefficient c that it produces is almost always a very good approximation to the actual value. Indeed, the RK4 method has a total accumulated error on the order of h⁴.

See also

• numerical ordinary differential equations
• midpoint method

References

• George E. Forsythe, Michael A. Malcolm, and Cleve B. Moler. Computer Methods for Mathematical Computations. Englewood Cliffs, NJ: Prentice-Hall, 1977. (See Chapter 6.)

• William H. Press, Brian P. Flannery, Saul A. Teukolsky, William T. Vetterling. Numerical Recipes in C. Cambridge, UK: Cambridge University Press, 1988. (See Sections 15.1 and 15.2.)