Midpoint method

In numerical analysis, a branch of applied mathematics, the midpoint method is a one-step method for numerically solving the differential equation,

The explicit midpoint method is given by the formula

the implicit midpoint method by

for ${\displaystyle n=0,1,2,\dots }$ Here, ${\displaystyle h}$ is the step size — a small positive number, ${\displaystyle t_{n}=t_{0}+nh,}$ and ${\displaystyle y_{n}}$ is the computed approximate value of ${\displaystyle y(t_{n}).}$ The explicit midpoint method is also known as the modified Euler method, the implicit method is the most simple collocation method, and, applied to Hamiltionian dynamics, a symplectic integrator.

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