Explicit amplification
Forward Euler repeatedly multiplies by one factor.
uₙ=(1+λΔt)ⁿu₀
Instability in Stiff Systems
When fast decay forces tiny explicit time steps
A stiff system contains fast-decaying modes that severely restrict explicit time stepping even after the physical transient has disappeared.
watchone idea
→manipulateone example
→leave withone intuition
The essential idea: for u′=λu with λ<0, Forward Euler is stable only when |1+λΔt|<1.
One Concept · One Example
Instability in Stiff Systems
Presented by Charlotte Moser · Coming soon
What to notice
- λ=−1000 gives the threshold Δt<0.002.
- At Δt=0.002, the amplitude does not decay.
- At Δt=0.0021, oscillations grow.
The mathematical structure
Implicit amplification
Backward Euler damps every negative real λ for any positive step.
uₙ₊₁=uₙ/(1−λΔt)
Cross the stability boundary
Reproduce the slides' λ=−1000 experiment and compare exact, Forward Euler, and Backward Euler solutions.
In stiff problems, numerical stability rather than physical time scale can dominate computational cost.