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Runge–Kutta (RK4)

The classic Runge–Kutta method (RK4) is a fourth-order method that uses four function evaluations per step to estimate the slope more accurately than Euler. Global error is $O(h^4)$ , so a small reduction in $h$ gives a large gain in accuracy.

y_{n+1} = y_n + \tfrac{h}{6}(k_1 + 2k_2 + 2k_3 + k_4)

the RK4 update

Symbols

k_1,\ldots,k_4

four slopes evaluated at different points within the step

Example

Dividing the step size by 10 in an order-2 method (Heun) makes the error roughly $10^2 = 100$ times smaller; for RK4 it becomes $10^4$ times smaller.

RK4 is more expensive per step than Euler (four evaluations versus one), but it almost always pays off in accuracy.

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