Runge-Kutta Methods are a family of [[Quadrature|Numerical Integration]] methods for solving systems of [[Differential Equations]]. ### Runge-Kutta 4 RK4 is the most common [[Runge-Kutta Methods|Runge-Kutta Method]]. Here is the [[Butcher Tableau]] of [[Runge-Kutta Methods|Runge-Kutta 4]]. $\huge \begin{array}{c|cccccc} t & k_{1}&k_{2}&k_{3}&k_{4} \\ \hline 0 \\ \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & 0 & \frac{1}{2} \\ 1 & 0 & 0 & 1 \\ \hline & \frac{1}{6} & \frac{1}{3} & \frac{1}{3} & \frac{1}{6} \end{array} $ $\large\begin{align} k_{1} &= f(t_{i},z_{i}) \\ k_{2} &= f\left( t_{i} + \frac{1}{2}h, z_{i} + \frac{1}{2}hk_{1} \right)\\ k_{3} &= f\pa{ t_{i} + \frac{1}{2}h, z_{i} + \frac{1}{2} hk_{2} } \\ k_{4} &= f\pa{t_{i} +h, z_{i} + hk_{3}} \\ \hline z_{i+1} &= z_{i} + h\pa{ \frac{1}{6}k_{1} + \frac{1}{3}k_{2} + \frac{1}{3}k_{3} + \frac{1}{6}k_{4} } \end{align} $