Runge-Kutta Method

• Simple Euler method comes from using just one term from the Taylor series for y(x) expanded about $x=x_0$.
• What if we use more terms of the Taylor series? Runge and Kutta, developed algorithms from using more than two terms of the series.
• In the Euler method, the increment is directly from $x_i$ to $x_{i+1}$.
• Second-order Runge-Kutta methods are obtained by using a weighted average of two increments to $y(x_0), k_1$ and $k_2$.
• Let's take a "trial step" in the middle and then increment to $x_{i+1}$ by using these middle x- and y-values. Two quantities are defined here as $k_1$ and $k_2$,
  

  $$k_1 = hf(x_i,y_i)$$

  $$k_2 = hf(x_i+\frac{1}{2}h,y_i+\frac{1}{2}k_1)$$

  
  

• The parameter;
  • $k_1$ is for the calculation at $x_i, y_i$,
  • $k_2$ is for a half-step away ( $x_i+\frac{1}{2}h,y_i+\frac{1}{2}k_1$) calculation.
• Accordingly, the 2$^{nd}$ order Runge-Kutta formula becomes:

  $$\boxed{y_{i+1}=y_i+k_2+O(h^3)}$$

• In the Runge-Kutta method, the margin of error in one step is $O(h^3)$ and is $O(h^2)$ in the entire interval.
• It works better than the Euler method, but it comes at a cost: f(x, y) will be calculated twice at each step.
• This "trial step" technique can be taken even further. Fourth-order Runge-Kutta (RK4) methods are most widely used and are derived in similar fashion.

• The local error term for the fourth-order Runge-Kutta method is $O(h^5)$; the global error would be $O(h^4)$.
• It is computationally more efficient than the (modified) Euler method because the steps can be manyfold larger for the same accuracy.
• However, four evaluations of the function are required per step rather than two.

• Let's apply the RK4 method on the previous example. Given differential equation,

  $$\frac{dy}{dx}=x+y$$

• The analytical solution of this equation is given as $y(x)=2e^x-x-1$. Initial condition: $y(x=0)=1$

Table 5.2: Solution of the differential equation $dy/dx=x+y$ in the interval [0, 1] by 4th order Runge-Kutta method.

(Example py-file: myrungekutta.py) As can be seen from the Table, much more sensitive results are obtained compared to the Euler method.

Figure 5.2: Solution of the differential equation $dy/dx=x+y$ in the interval [0, 1] by Euler method.