Midpoint method
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In numerical analysis, a branch of applied mathematics, the midpoint method is a one-step method for solving the differential equation
numerically, and is given by the formula
Here, h is the step size — a small positive number, tn = t0 + nh, and yn is the computed approximate value of y(tn).
The name of the method comes from the fact that tn + h / 2 is the midpoint between tn at which the value of y(t) is known and tn + 1 at which the value of y(t) needs to be found.
The method is an example of a class of methods known as Runge-Kutta methods.
[edit] Derivation of the midpoint method
The midpoint method is a refinement of the Euler's method
and is derived in a similar manner. The key to deriving Euler's method is the approximate equality
which is obtained from the slope formula
and keeping in mind that y' = f(t,y).
For the midpoint method, one replaces (3) with the more accurate
when instead of (2) we find
One cannot use this equation to find y(t + h) as one does not know y at t + h / 2. The solution is then to use a Taylor series expansion
which, when plugged in (4), gives us
and the midpoint method (1).
