Topic 14.4: Multiple-Step Methods (Theory)

Given the IVP

y⁽¹⁾(t) = f( t, y(t) )
y(a) = y₀

where we want to estimated y(b) for b > a, we can apply the same strategy we used for the composite-trapezoidal rule:

Break the interval [a, b] into n sub-intervals and define h = (b - a)/n. Then set t_i = a + ih for i = 0, 1, ..., n. Therefore t₀ = a and t_n = b.

Now, y₀ is the initial value so let y_i represent the approximation of y(t_i) for i = 1, 2, ..., n.

Using any of the three techniques we've seen, Euler's, Heun's, or 4th-order Runge Kutta, we may now proceed as follows:

For i from 1 to n:

Approximate y_i using y_{i − 1}.

More specifically:

Calculate

y_i = y_{i − 1} + h f(t_{i − 1}, y_{i − 1})

for i = 1, 2, ..., n.

Calculate

K₀ = f(t_{i − 1}, y_{i − 1})
K₁ = f(t_i, y_{i − 1} + h K₀)

and set y_i = y_{i − 1} + h (K₀ + K₁)/2.

Calculate

K₀ = f(t_{i − 1}, y_{i − 1})
K₁ = f(t_{i − 1} + ½h, y_{i − 1} + ½h K₀)
K₂ = f(t_{i − 1} + ½h, y_{i − 1} + ½h K₁)
K₃ = f(t_i, y_{i − 1} + h K₂)

and set y_i = y_{i − 1} + h (K₀ + 2 K₁ + 2 K₂ + K₃)/6.