midpoint method

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

y'(t) = f(t, y(t)), \quad y(t_0) = y_0

numerically, and is given by the formula

y_{n+1} = y_n + hf\left(t_n+\frac{h}{2},y_n+\frac{h}{2}f(t_n, y_n)\right), \ n=0, 1, 2, \dots  \qquad\qquad (1)

Here,

h

is the step size — a small positive number,

t_n=t_0 + n h,

and

y_n

is the computed approximate value of

y(t_n).

The name of the method comes from the fact that

t_n+h/2

is the midpoint between

t_n

at which the value of y(t) is known and

t_{n+1}

at which the value of y(t) needs to be found.

Derivation of the midpoint method

The midpoint method is a refinement of the Euler's method

y_{n+1} = y_n + hf(t_n,y_n),\,

and is derived in a similar manner. The key to deriving Euler's method is the approximate equality

y(t+h) \approx y(t) + hf(t,y(t)) \qquad\qquad (2)

which is obtained from the slope formula

y'(t) \approx \frac{y(t+h) - y(t)}{h} \qquad\qquad (3)

and keeping in mind that

y' = f(t, y).

For the midpoint method, one replaces (3) with the more accurate

y'\left(t+\frac{h}{2}\right) \approx \frac{y(t+h) - y(t)}{h}

when instead of (2) we find

y(t+h) \approx y(t) + hf\left(t+\frac{h}{2},y\left(t+\frac{h}{2}\right)\right). \qquad\qquad (4)

One cannot use this equation to find

y(t+h)

as one does not know

y

t+h/2.

The solution is then to use a Taylor series expansion

y\left(t + \frac{h}{2}\right) \approx y(t) + \frac{h}{2}y'(t)=y(t) + \frac{h}{2}f(t, y(t)),

which, when plugged in (4), gives us

y(t + h) \approx y(t) + hf\left(t + \frac{h}{2}, y(t) + \frac{h}{2}f(t, y(t))\right)

and the midpoint method (1).