Trapezoidal rule (differential equations)

The trapezoidal rule for differential equations is a numerical method for solving ordinary differential equations of the form y prime equals f of t and y. It borrows its name and its idea from the trapezoidal rule for computing integrals: each step replaces the integral of the derivative over one time step by the average of the derivative at the two ends of the step, multiplied by the step size. Because the unknown value at the end of the step appears on both sides of the formula, the method is implicit, and every step requires solving an equation that is usually nonlinear, for example with Newton's method started from an Euler estimate. It is a second-order method that can be viewed both as a Runge-Kutta method and as a linear multistep method. Its region of absolute stability is exactly the left half of the complex plane, so it is A-stable, and the second Dahlquist barrier says no A-stable linear multistep method is more accurate.