Comparison Explicit Euler / Implicit Euler
Explicit Euler
- - \(u^{k+1}=u^{k}+h\,\mathcal{F}(u^{k},t^{k})\)
- - \(u^{k+1}=(\mathrm{I}+h\,\mathrm{A})\,u^{k}+h\,b(t^k)\)
- - Stability:
-
- Conditional \(|1+h\lambda|\geq 1\)
- Solution to spring model diverges
Implicit Euler
- - \(u^{k+1}=u^{k}+h\,\mathcal{F}(u^{k+1},t^{k+1})\)
- - \(u^{k+1}=(\mathrm{I}-h\,\mathrm{A})^{-1}\left(\,u^{k}+h\,b(t^{k+1})\right)\)
- - Stability:
-
- Unconditionnaly stable
- Solution to spring model converge to \(l^0\)