Explicit Euler - study case
Free fall under gravity
- - True solution \(\tilde u(k\,h)=p_0 + (k\,h) v_0 + \frac{(k\,h)^2}{2}\,g\)
- - Numerical scheme: \(\left(\begin{array}{c}p \\ v\end{array}\right)^{k+1} = \left(\begin{array}{cc}1 & h \\ 0 & 1
\end{array}\right) \left(\begin{array}{c}p \\ v\end{array}\right)^k+\left(\begin{array}{c}0 \\ h\,g\end{array}\right)\)
- - Numerical solution: \(p^k= p_0 + k\,h\, v_0 + \frac{k (k-1)}{2}\,h^2\,g \)
\(\Rightarrow\) Not exact : Error \(e^k = |u^k-\tilde u(k\,h)| = \frac{g}{2} k\,h^2 \)
- red: true solution
- blue: numerical solution