Midpoint method explained

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)), y(t₀₎=y₀.

The explicit midpoint method is given by the formulathe implicit midpoint method byfor

n=0,1,2,...

Here,

is the step size - a small positive number,

t_n=t₀+nh,

and

y_n

is the computed approximate value of

y(t_n).

The explicit midpoint method is sometimes also known as the modified Euler method, the implicit method is the most simple collocation method, and, applied to Hamiltonian dynamics, a symplectic integrator. Note that the modified Euler method can refer to Heun's method, for further clarity see List of Runge–Kutta methods.

The name of the method comes from the fact that in the formula above, the function

giving the slope of the solution is evaluated at

t=t_n+h/2=\tfrac{t_n+t_n+1

}, 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.

A geometric interpretation may give a better intuitive understanding of the method (see figure at right). In the basic Euler's method, the tangent of the curve at

(t_n,y_n)

is computed using

f(t_n,y_n)

. The next value

y_n+1

is found where the tangent intersects the vertical line

t=t_n+1

. However, if the second derivative is only positive between

t_n

and

t_n+1

, or only negative (as in the diagram), the curve will increasingly veer away from the tangent, leading to larger errors as

increases. The diagram illustrates that the tangent at the midpoint (upper, green line segment) would most likely give a more accurate approximation of the curve in that interval. However, this midpoint tangent could not be accurately calculated because we do not know the curve (that is what is to be calculated). Instead, this tangent is estimated by using the original Euler's method to estimate the value of

y(t)

at the midpoint, then computing the slope of the tangent with

. Finally, the improved tangent is used to calculate the value of

y_n+1

from

y_n

. This last step is represented by the red chord in the diagram. Note that the red chord is not exactly parallel to the green segment (the true tangent), due to the error in estimating the value of

y(t)

at the midpoint.

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\to0

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

Derivation of the midpoint method

The midpoint method is a refinement of the Euler 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 equalitywhich is obtained from the slope formulaand keeping in mind that

y'=f(t,y).

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

y'\left(t+	h
	2

\right) ≈

	y(t+h)-y(t)
	h

when instead of (2) we find

One cannot use this equation to find

y(t+h)

as one does not know

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+

	h
	2

\right) ≈ y(t)+

	h
	2

y'(t)=y(t)+

	h
	2

f(t,y(t)),

which, when plugged in (4), gives us

y(t+h) ≈ y(t)+hf\left(t+

	h
	2

,y(t)+

	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)

y(t+h)

y\left(t+

h2\right) ≈

	12l(y(t)+y(t+h)r)

and thus

	y(t+h)-y(t)
	h

≈ y'\left(t+

h2\right) ≈

k=f\left(t+

h2,	12l(y(t)+y(t+h)r)\right)

Inserting the approximation

y_n+hk

for

y(t_n+h)

results in the implicit Runge-Kutta method

\begin{align} k&=f\left(t

h2,y

n+	h2
	k\right)\\ y

_n+1&=y_{n+hk
\end{align}}

which contains the implicit Euler method with step size

h/2

as its first part.

Because of the time symmetry of the implicit method, allterms of even degree in

of the local error cancel, so that the local error is automatically of order

lO(h³⁾

. Replacing the implicit with the explicit Euler method in the determination of

results again in the explicit midpoint method.

References

Book: Griffiths, D. V. . Smith, I. M. . Numerical methods for engineers: a programming approach. CRC Press. Boca Raton. 1991. 0-8493-8610-1. 218.
.
Book: Burden . Richard . Faires . John . Numerical Analysis . Richard Stratton. 2010. 978-0-538-73351-9. 286.

Midpoint method explained

Derivation of the midpoint method

See also

References