RK2 midpoint
A midpoint slope corrects the Euler direction.
k₁=f(uₙ,tₙ), k₂=f(uₙ+Δtk₁/2,tₙ+Δt/2)
Higher-Order Numerical Methods
Improving accuracy and efficiency in time integration
Higher-order schemes combine more information per step to achieve faster convergence than Forward Euler.
watchone idea
→manipulateone example
→leave withone intuition
The essential idea: RK2 and Crank-Nicolson are second order, while classical RK4 is fourth order for smooth non-stiff problems.
One Concept · One Example
Higher-Order Numerical Methods
Presented by Charlotte Moser · Coming soon
What to notice
- Runge-Kutta methods sample intermediate slopes.
- Multistep and implicit methods use different information.
- Order predicts how error changes with Δt.
The mathematical structure
RK4
Four weighted slopes produce fourth-order accuracy.
uₙ₊₁=uₙ+(Δt/6)(k₁+2k₂+2k₃+k₄)
Compare four numerical schemes
Use the slides' equation u′=−2u+sin(t), u(0)=1. Compare Forward Euler, RK2, RK4, and Crank-Nicolson.
Higher-order methods spend more work per step to reduce error much faster as the grid is refined.