Implicit Euler - study case
1D spring
- - True solution: permanent oscillations
- - Numerical scheme: \(\left(\begin{array}{c}p \\ v\end{array}\right)^{k+1} = \left(\begin{array}{cc}1 & -h \\ K/m\,h & 1
\end{array}\right)^{-1}\left(\left(\begin{array}{c}p \\ v\end{array}\right)^k+\left(\begin{array}{c}0 \\ K/m\,h\,l^0\end{array}\right)\right)\)
- \(\Rightarrow \left(\begin{array}{c}p \\ v\end{array}\right)^{k+1} = \frac{1}{1+h^2\,K/m}\left(\left(\begin{array}{cc}1 & h \\ -K/m\,h & 1
\end{array}\right)\left(\begin{array}{c}p \\ v\end{array}\right)^k+\left(\begin{array}{c}K/m\,h^2\,l^0 \\ K/m\,h\,l^0\end{array}\right)\right)\)
- Eigenvalues of \((\mathrm{I}-\mathrm{A}h)^{-1}\) are \(\frac{1\pm i\sqrt{\frac{K}{m}}h}{1+\frac{K}{m}h^2}\)
- \(\Rightarrow \left|\frac{1\pm i\sqrt{\frac{K}{m}}h}{1+\frac{K}{m}h^2} \right| =\frac{1}{\sqrt{1+\frac{K}{m}h^2}} < 1\) \(\Rightarrow\) always
converge
- Even if the true solution oscillates
