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The Mid-Point Method

One of the remedies to the problem that we have with the Euler method is to sample more frequently. Consider, for example, the following scheme:

k1 = $\displaystyle \Delta t f\left(x_n, t_n\right)$  
k2 = $\displaystyle \Delta t f\left(x_n + \frac{1}{2} k_1, t_n + \frac{1}{2}\Delta t\right)$  
xn+1 = $\displaystyle x_n + k_2 + {\cal O}\left((\Delta t)^3\right)$  

We can combine all those three equations together to obtain the following expression:

\begin{displaymath}x_{n+1} = x_n +
f\left(x_n + f\left(x_n, t_n\right) \frac{1}{2}\Delta t,
t_n + \frac{1}{2}\Delta t\right)\Delta t
\end{displaymath} (4.77)

In other words, in this scheme we make a simple Euler step to the mid-point of the interval $[t_n, t_n + \Delta t]$, find the corresponding xn + 1/2 and evaluate $f\left(x_{n + 1/2}, t_{n+1/2}\right)$, and then use that value in order to jump from tn to tn+1.

The mid-point method is second-order accurate.

The mid-point method is the simplest example of the family of Runge-Kutta methods. It is also referred to as the second-order Runge-Kutta method.


next up previous index
Next: The Fourth-Order Runge-Kutta Method Up: Pushing Particles Previous: Euler Method
Zdzislaw Meglicki
2001-02-26