Implicit Euler - study case

Free fall under gravity

• - True solution \(\tilde u(k\,\Delta t)=p_0 + (k\,\Delta t) v_0 + \frac{(k\,\Delta t)^2}{2}\,g\)

• - Numerical scheme: \(\left(\begin{array}{c}p \\ v\end{array}\right)^{k+1} = \left(\begin{array}{cc}1 & -h \\ 0 & 1 \end{array}\right)^{-1}\left(\left(\begin{array}{c}p \\ v\end{array}\right)^k+\left(\begin{array}{c}0 \\ g\end{array}\right)\right)\)

• \(\Rightarrow \left(\begin{array}{c}p \\ v\end{array}\right)^{k+1} = \left(\begin{array}{cc}1 & h \\ 0 & 1 \end{array}\right)\left(\begin{array}{c}p \\ v\end{array}\right)^k+\left(\begin{array}{c}h^2\, g \\ h\,g\end{array}\right)\)

• - Numerical solution: \(p^k= p_0 + (k\,\Delta t) v_0 + \frac{k (k+1)}{2}\,(\Delta t)^2\,g \)

• Same error magnitude than explicit Euler

• - red: True solution
• - magenta: Implicit Euler
• - blue: Explicit Euler