Midpoint method

The midpoint method is a one-step numerical method for solving an ordinary differential equation that is given together with an initial value. It improves on the basic Euler method by estimating the slope of the solution at the middle of each step instead of at its start. In the explicit form, an Euler half step first estimates the solution at the midpoint of the interval, the slope is evaluated there, and that slope is used to take the full step. The explicit version is sometimes called the modified Euler method, a name that can also refer to Heun's method. The implicit form is the simplest collocation method and, applied to Hamiltonian dynamics, a symplectic integrator. The local error of each step is of order h cubed, where h is the step size, so the method costs more per step than Euler's method but its error generally shrinks faster as the step size falls. It belongs to the family of Runge-Kutta methods.