Update rule
Replace the differential change by a finite increment.
uₙ₊₁=uₙ+Δt f(uₙ,tₙ)
Forward Euler Method
Approximating continuous dynamics by discrete steps
Forward Euler advances an ODE by following the current slope over one finite time step.
watchone idea
→manipulateone example
→leave withone intuition
The essential idea: the update is simple and explicit, but its global error decreases only linearly with Δt.
One Concept · One Example
Forward Euler Method
Presented by Charlotte Moser · Coming soon
What to notice
- Each step uses f at the current state.
- Smaller Δt improves accuracy.
- Stability may impose a stricter limit than accuracy.
The mathematical structure
First-order convergence
Halving Δt should roughly halve the global error.
||u_exact(T)−u_num(T)||≤CΔt
Build a numerical trajectory step by step
Apply Forward Euler to u′=−u and compare its straight-line updates with the exact exponential solution.
Forward Euler turns a local slope into a full approximation, with error proportional to the step size.