$$\newcommand{\D}{\displaystyle} \renewcommand{\eqref}{Eq.~(\ref{#1})}$$

# 2.11 Runge-Kutta of 4th order

Euler's method and Heun's method belong to the Runge-Kutta family of explicit methods, and is respectively Runge-Kutta of 1st and 2nd order, the latter with one time use of corrector. Explicit Runge-Kutta schemes are single step schemes that try to copy the Taylor series expansion of the differential equation to a given order.

The classical Runge-Kutta scheme of 4th order (RK4) is given by \begin{align} &k_1=f(x_n,y_n)\nonumber\\ &k_2=f(x_n+\frac{h}{2}, y_n+\frac{h}{2}k_1)\nonumber\\ \tag{2.96} &k_3=f(x_n+\frac{h}{2},y_n+\frac{h}{2}k_2)\\ &k_4=f(x_n+h,y_n+hk_3)\nonumber\\ &y_{n+1}=y_n+\frac{h}{6}(k_1+2k_2+2k_3+k_4)\nonumber \end{align}

We see that we are actually using Euler's method four times and find a weighted gradient. The local error is of order $$O(h^5)$$, while the global is of $$O(h^4)$$. We refer to .

Figure 15 shows a graphical illustration of the RK4 scheme.

In detail we have

1. In point $$(x_n,y_n)$$ we know the gradient $$k_1$$ and use this when we go forward a step $$h/2$$ where the gradient $$k_2$$ is calculated.
2. With this gradient we start again in point $$(x_n,y_n)$$, go forward a step $$h/2$$ and find a new gradient $$k_3$$.
3. With this gradient we start again in point $$(x_n,y_n)$$, but go forward a complete step $$h$$ and find a new gradient $$k_4$$.
4. The four gradients are averaged with weights $$1/6$$, $$2/6$$, $$2/6$$ and $$1/6$$. Using the averaged gradient we calculate the final value $$y_{n+1}$$.
Each of the steps above are Euler steps.

Figure 15: Illustration of the RK4 scheme. Using (2.96) on the equation system in (2.64) we get \begin{align} &(y_1)_{n+1}=(y_1)_n +\frac{h}{6}(k_1+2k_2+2k_3+k_4) \nonumber\\ &(y_2)_{n+1}=(y_2)_n +\frac{h}{6}(l_1+2l_2+2l_3+l_4) \tag{2.97} \\ &(y_3)_{n+1}=(y_3)_n +\frac{h}{6}(m_1+2m_2+2m_3+m_4) \nonumber\\ \tag{2.98} \end{align} where \begin{align*} k_1&=y_2 \\ l_1&=y_3 \\ m_1&=-y_1y_3\\ \\ k_2&=(y_2+hl_l/2)\\ l_2&=(y_3+hm_1/2)\\ m_2&=-[(y_1+hk_1/2)(y_3+hm_1/2)]\\ \\ k_3&=(y_2+hl_2/2)\\ l_3&=(y_3+hm_2/2)\\ m_3&=-[(y_1+hk_2/2)(y_3+hm_2/2)]\\ \\ k_4&=(y_2+hl_3)\\ l_4&=(y_3+hm_3)\\ m_4&=-[(y_1+hk_3)(y_3+hm_3) \end{align*}