Wikipedia / Midpoint method
Wikipedia / Midpoint method
In numerical analysis, a branch of applied mathematics, the midpoint method is a one-step method for numerically solving the differential equation,
The explicit midpoint method is given by the formula the implicit midpoint method by is the step size 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.
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 at the midpoint.
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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
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
For the midpoint methods, one replaces (3) with the more accurate
when instead of (2) we find
. The solution is then to use a Taylor series expansion exactly as if using the Euler method:
which, when plugged in (4), gives us
and the explicit midpoint method (1e).
and thus
results in the implicit Runge-Kutta method
as its first part.
results again in the explicit midpoint method.
See also
Notes
References
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