Mid-Point Method

Objective: Want to evaluate $\dot{x}=f(x)$ between $t_k$ and $t_{k+1}$ (omitting $u$ and $t$ for ease of notation):

$$x_{k+1}=x_k+\Delta tf\left(x\left(t_k+\frac{1}{2}\Delta t\right)\right)$$

The problem is that we don’t know what $x\left(t_k+\frac{1}{2}\Delta t\right)$ is. However, it can be approximated by using the Euler Half-Step

We evaluate the function in the mid-point

Accuracy

One-step error is order $\mathcal{O}(\Delta t^3)$

• Due to extra taylor expansion part, compared to Accuracy of the Forward (Explicit) Euler Method Global error is order $\mathcal{O}(\Delta t^2)$