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

for  Here,  is the step size — a small positive number,  and  is the computed approximate value of  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  the midpoint between  at which the value of  is known and  at which the value of  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  is computed using . The next value  is found where the tangent intersects the vertical line . However, if the second derivative is only positive between  and , 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  at the midpoint, then computing the slope of the tangent with . Finally, the improved tangent is used to calculate the value of  from . 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  at the midpoint. 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's error generally decreases faster as . 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

and keeping in mind that 

For the midpoint methods, 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 explicit midpoint method (1e). The implicit method (1i) is obtained by approximating the value at the half step  by the midpoint of the line segment from  to 

and thus
 
Inserting the approximation  for 
results in the implicit Runge-Kutta method

which contains the implicit Euler method with step size  as its first part. Because of the time symmetry of the implicit method, all
terms of even degree in  of the local error cancel, so that the local error is automatically of order . Replacing the implicit with the explicit Euler method in the determination of  results again in the explicit midpoint method. See also
 Rectangle method
 Heun's method
 Leapfrog integration and Verlet integration

Notes

References

 
 . Numerical differential equations
Runge–Kutta methods