4.4: Midpoint method
- Page ID
- 219161
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A similar method to Heun’s is the midpoint method. We will specifically look at the explicit midpoint method (there is also an implicit midpoint method). The midpoint method uses forward Euler to take a half-step forward, then computes the slope at that point and uses that slope to make the full step. The algorithm is presented in [alg:5]. A drawing showing how the midpoint method works is shown in 4.6.
Inputs: initial condition \(y_0\), end time \(t_{end}\).
Initialize: \(t_0 = 0, y_0\).
for \(n=0: t_{end}/h\) do
Compute slope at \(t_n\): \(s_1 \leftarrow f(t_n, y_n)\)
Compute intermediate step to \(t_{n} + h/2\): \(\tilde{y} \leftarrow y_n + (h/2) s_1\)
Compute slope at \(t_{n} + h/2\): \(s_2 \leftarrow f(t_{n}+h/2, \tilde{y})\)
Compute final step: \(y_{n+1} \leftarrow y_n + h s_2\)
end for
return \(y\) vector.
