Heun's method

First compute a step using Euler's method. Denote it with a capital letter to distinguish it from the result of Heun's method.

X_i^m+1 = x_i^m + h v_i^m

V_i^m+1 = v_i^m + h a_i^m

Heun's method proceeds by averaging the "slopes" used at the current time m and the predicted values from the Euler step.

x_i^m+1 = x_i^m + h/2 (v_i^m + V_i^m+1)

v_i^m+1 = v_i^m + h/2 (a_i^m + A_i^m+1)

The acceleration A_i^m+1 is calculated from the usual acceleration formula using the Euler-predicted positions X_i^m+1.

This method is like the trapezoidal rule but with a predicted Euler step in place of the unknown values.

in the Burden and Faires "Numerical Analysis" textbook: page 277 in Section 5.4 (as the "modified Euler" method)
in Hairer, Norsett, and Wanner's "Solving Ordinary Differential Equations I: Nonstiff Problems"
HNMI at USF
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