Numerical Methods Explorer
| Method | h = 0.5 | h = 0.25 | h = 0.125 | Ratio (0.5/0.25) | Order |
|---|
Euler — O(h)
X_{n+1} = X_n
+ h·f(t_n, X_n)
+ h·f(t_n, X_n)
1 function evaluation per step. Local error O(h²), global O(h). Halving h ≈ halves the error.
Modified Euler (RK2) — O(h²)
X̃ = X_n + h·f(t_n, X_n)
X_{n+1} = X_n + h/2·[f(t_n, X_n)
+ f(t_{n+1}, X̃)]
X_{n+1} = X_n + h/2·[f(t_n, X_n)
+ f(t_{n+1}, X̃)]
Predictor–corrector. 2 evaluations per step. Halving h ≈ quarters the error.
RK4 — O(h⁴)
k₁=f(tₙ,Xₙ)
k₂=f(tₙ+h/2, Xₙ+h/2·k₁)
k₃=f(tₙ+h/2, Xₙ+h/2·k₂)
k₄=f(tₙ+h, Xₙ+h·k₃)
X_{n+1}=Xₙ+h/6·(k₁+2k₂+2k₃+k₄)
k₂=f(tₙ+h/2, Xₙ+h/2·k₁)
k₃=f(tₙ+h/2, Xₙ+h/2·k₂)
k₄=f(tₙ+h, Xₙ+h·k₃)
X_{n+1}=Xₙ+h/6·(k₁+2k₂+2k₃+k₄)
4 evaluations per step. Halving h reduces error by factor ≈ 16.
Error at t = T for current h