Midpoint method

For the midpoint rule in numerical quadrature, see rectangle method.

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,

$y'(t)=f(t,y(t)),\quad y(t_{0})=y_{0}$.

The explicit midpoint method is given by the formula

$y_{n+1}=y_{n}+hf\left(t_{n}+{\frac {h}{2}},y_{n}+{\frac {h}{2}}f(t_{n},y_{n})\right),\qquad \qquad (1e)$

the implicit midpoint method by

$y_{n+1}=y_{n}+hf\left(t_{n}+{\frac {h}{2}},{\frac {1}{2}}(y_{n}+y_{n+1})\right),\qquad \qquad (1i)$

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 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.

The name of the method comes from the fact that in the formula above the function $f$ giving the slope of the solution is evaluated at $t=t_{n}+h/2,$ (note : $t$ is f ${\frac {t_{n}+t_{n+1}}{2}}$) 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's error generally decreases faster as $h\to 0$.

The methods are examples of a class of higher-order methods known as Runge-Kutta methods.

Derivation of the midpoint method

Illustration of numerical integration for the equation $y'=y,y(0)=1.$ 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

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

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

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

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 explicit midpoint method (1e).

The implicit method (1i) is obtained by approximating the value at the half step $t+h/2$ by the midpoint of the line segment from $y(t)$ to $y(t+h)$

$y\left(t+{\frac {h}{2}}\right)\approx {\frac {1}{2}}{\bigl (}y(t)+y(t+h){\bigr )}$

and thus

${\frac {y(t+h)-y(t)}{h}}\approx y'\left(t+{\frac {h}{2}}\right)\approx k=f\left(t+{\frac {h}{2}},{\frac {1}{2}}{\bigl (}y(t)+y(t+h){\bigr )}\right)$

Inserting the approximation $y_{n}+h\,k$ for $y(t_{n}+h)$ results in the implicit Runge-Kutta method

${\begin{aligned}k&=f\left(t_{n}+{\frac {h}{2}},y_{n}+{\frac {h}{2}}k\right)\\y_{n+1}&=y_{n}+h\,k\end{aligned}}$

which contains the implicit Euler method with step size $h/2$ as its first part.

Because of the time symmetry of the implicit method, all terms of even degree in $h$ of the local error cancel, so that the local error is automatically of order ${\mathcal {O}}(h^{3})$. Replacing the implicit with the explicit Euler method in the determination of $k$ results again in the explicit midpoint method.

See also

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

References

• Griffiths,D. V.; Smith, I. M. (1991). Numerical methods for engineers: a programming approach. Boca Raton: CRC Press. p. 218. ISBN 0-8493-8610-1.
• Süli, Endre; Mayers, David (2003), An Introduction to Numerical Analysis, Cambridge University Press, ISBN 0-521-00794-1.