Higher order method - RK

Higher accuracy can be achived using Taylor expansion.

- ex. \(\displaystyle u^{k+1}=u^{k}+h\,\mathcal{F}(u^k,t^k)+\frac{h^2}{2}\frac{\mathrm{d}\mathcal{F}}{\mathrm{d}t}(u^k,t^k)\)

- Second order accurate: 2nd order explicit Euler

- Higher order can achieve arbitrary accuracy.
- In practice: computing derivatives of \(\mathcal{F}\) is complex.
\(\Rightarrow\) Not often used in practice

Instead: Runge Kutta
- - Reach higher order accuracy
- - Only involve the knowledge of \(\mathcal{F}\), without its derivatives
- - Involves several successives evaluations of \(\mathcal{F}\)