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

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, 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

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 at 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).

[edit] See also

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