Mr. Integral
Math notes/Differential Equations/Numerical Methods

Midpoint, Heun and Runge–Kutta Methods — Cheat sheet

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Core rule

yn+1=yn+hf(tn+h/2,yn+(h/2)f(tn,yn)).y_{n+1}=y_n+h f(t_n+h/2,y_n+(h/2)f(t_n,y_n)).

Watch for

Higher order does not remove stability limits or repair discontinuous forcing automatically. Split a step at known discontinuities. Adaptive methods estimate local error and adjust steps, but the requested tolerance is not an unconditional bound on every global error.