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In mathematics of stochastic systems, the Runge–Kutta method is a technique for the approximate numerical solution of a stochastic differential equation.It is a generalisation of the Runge–Kutta method for ordinary differential equations to stochastic differential equations (SDEs).
The Runge–Kutta–Fehlberg method has two methods of orders 5 and 4; it is sometimes dubbed RKF45 . Its extended Butcher Tableau is: / / / / / / / / / / / / / / / / / / / / / / / / / / The first row of b coefficients gives the fifth-order accurate solution, and the second row has order four.
The consequence of this difference is that at every step, a system of algebraic equations has to be solved. This increases the computational cost considerably. If a method with s stages is used to solve a differential equation with m components, then the system of algebraic equations has ms components.
If , then the step is completed. Replace h {\textstyle h} with h new {\textstyle h_{\text{new}}} for the next step. The coefficients found by Fehlberg for Formula 2 (derivation with his parameter α 2 = 3/8) are given in the table below, using array indexing of base 1 instead of base 0 to be compatible with most computer languages:
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