Trapezoidal rule (differential equations)

In numerical analysis and scientific computing, the trapezoidal rule is a numerical method to solve ordinary differential equations derived from the trapezoidal rule for computing integrals. The trapezoidal rule is an implicit second-order method, which can be considered as both a Runge–Kutta method and a linear multistep method.

Suppose that we want to solve the differential equation

The trapezoidal rule is given by the formula

where ${\displaystyle h=t_{n+1}-t_{n}}$ is the step size.

This is an implicit method: the value ${\displaystyle y_{n+1}}$ appears on both sides of the equation, and to actually calculate it, we have to solve an equation which will usually be nonlinear. One possible method for solving this equation is Newton's method. We can use the Euler method to get a fairly good estimate for the solution, which can be used as the initial guess of Newton's method.

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