Definition

The midpoint method corresponds to the following choice of coefficients:

• a₁ = 0
• a₂ = 1
• a₃ = 1/2
• a₄ = f(t_n, y_n)/2

By inserting these coefficients into the Runge–Kutta form, we obtain the classical midpoint algorithm.

Midpoint Method Algorithm

Given: a time step h, an initial condition (t₀, y₀), and a maximum number of iterations N.

For 0 ≤ n ≤ N:

k₁ = h f(tₙ, yₙ)

yₙ₊₁ = yₙ + h · f(tₙ + h/2 ,  yₙ + k₁/2)

tₙ₊₁ = tₙ + h
            

Remarks

• Why it is called the midpoint method:
The function f(t, y) is evaluated at the midpoint of the interval [tₙ , tₙ₊₁]. This midpoint evaluation increases accuracy and leads to a second-order method.

• Precision:
The modified Euler method and the midpoint method have the same local truncation order. Therefore, both methods provide comparable accuracy, although their evaluation strategies differ.

• Other choices:
Several other choices of coefficients aᵢ are possible and lead to different second-order Runge–Kutta schemes. In this course, we focus only on these two classical variations.