Midpoint method

Article summary

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 the implicit midpoint method by for n=0, 1, 2, \dots Here, h is the step size — a small positive number, t_n=t_0 + n h, 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 f giving the slope of the solution is evaluated at t = t_n + h/2= \tfrac{t_n+t_{n+1}}{2}, 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} .…

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