Force density

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Illustration of the midpoint method assuming that $y_n$ equals the exact value $y(t_n).$ The midpoint method computes $y_{n+1}$ so that the red chord is approximately parallel to the tangent line at the midpoint (the green line).

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

$y^{\prime }(t)=f(t,y(t)),y(t_0)=y_0$

and is given by the formula

$y_{n+1}=y_n+hf(t_n+\frac{h}{2},y_n+\frac{h}{2}f(t_n,y_n)),(1)$

for $n=0,1,2,\dots$ Here, $h$ is the step size — a small positive number, $t_n=t_0+nh,$ and $y_n$ is the computed approximate value of $y(t_n).$ The midpoint method is also known as the modified Euler method.

The name of the method comes from the fact that in the formula above the function $f$ is evaluated at $t=t_n+h/2,$ which 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.

The local error at each step of the midpoint method is of order $O\left(h^3\right)$, giving a global error of order $O\left(h^2\right)$. Thus, while more computationally intensive than Euler's method, the midpoint method generally gives more accurate results.

The method is an example of a class of higher-order methods known as Runge-Kutta methods.

Derivation of the midpoint method

Illustration of numerical integration for the equation $y^{\prime }=y,y(0)=1.$ Blue: the Euler method, green: the midpoint method, red: the exact solution, $y=e^t.$ The step size is $h=1.0.$

The same illustration for $h=0.25.$ It is seen that the midpoint method converges faster than the Euler 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

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

which is obtained from the slope formula

$y^{\prime }(t)\approx \frac{y(t+h)-y(t)}{h}(3)$

and keeping in mind that $y^{\prime }=f(t,y).$

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

$y^{\prime }(t+\frac{h}{2})\approx \frac{y(t+h)-y(t)}{h}$

when instead of (2) we find

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

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 exactly as if using the Euler method to solve for $y(t+h/2)$:

$y(t+\frac{h}{2})\approx y(t)+\frac{h}{2}{y}^{\prime }(t)=y(t)+\frac{h}{2}f(t,y(t)),$

which, when plugged in (4), gives us

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

and the midpoint method (1).

See also

• Rectangle method
• Heun's method
• Leapfrog integration and Verlet integration

Notes

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