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Runge-Kutta Methods

Undergraduate

Definition

Runge-Kutta methods are high-precision techniques for numerical solutions of differential equations. Fourth-order (RK4) is most widely used.

k₁ = f(tₙ, yₙ)

First slope

k₂ = f(tₙ + (h)/(2), yₙ + (h)/(2)k₁)

Second slope

k₃ = f(tₙ + (h)/(2), yₙ + (h)/(2)k₂)

Third slope

k₄ = f(tₙ + h, yₙ + hk₃)

Fourth slope

y_n+1 = yₙ + (h)/(6)(k₁ + 2k₂ + 2k₃ + k₄)

RK4 update formula

Example 1

Apply RK4 to dy/dt = y, y(0) = 1 with h = 0.1.

Discovered by: Carl Runge, Martin Kutta (1901)

Developed combining multiple slopes to improve Euler method accuracy.

Orbital mechanics, particle motion

Physics engines

Control system simulation