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Illustration of the midpoint method assuming that equals the exact value The midpoint method computes 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,

and is given by the formula

for Here, is the step size — a small positive number, and is the computed approximate value of The midpoint method is also known as the modified Euler method.[1]

The name of the method comes from the fact that in the formula above the function is evaluated at which is the midpoint between at which the value of y(t) is known and at which the value of y(t) needs to be found.

The local error at each step of the midpoint method is of order , giving a global error of order . 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 Blue: the Euler method, green: the midpoint method, red: the exact solution, The step size is
The same illustration for It is seen that the midpoint method converges faster than the Euler method.

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

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

which is obtained from the slope formula

and keeping in mind that

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

when instead of (2) we find

One cannot use this equation to find as one does not know at The solution is then to use a Taylor series expansion exactly as if using the Euler method to solve for :

which, when plugged in (4), gives us

and the midpoint method (1).

See also

Notes

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References

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