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. Method 

Suppose that we want to solve the differential equation

The trapezoidal rule is given by the formula

where  is the step size. This is an implicit method: the value  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. Cutting short, using only the guess from Eulers method is equivalent to performing Heun's method. Motivation 

Integrating the differential equation from  to , we find that

The trapezoidal rule states that the integral on the right-hand side can be approximated as

Now combine both formulas and use that  and  to get the trapezoidal rule for solving ordinary differential equations. Error analysis 

It follows from the error analysis of the trapezoidal rule for quadrature that the local truncation error  of the trapezoidal rule for solving differential equations can be bounded as:

Thus, the trapezoidal rule is a second-order method. This result can be used to show that the global error is  as the step size  tends to zero (see big O notation for the meaning of this). Stability 

The region of absolute stability for the trapezoidal rule is

This includes the left-half plane, so the trapezoidal rule is A-stable. The second Dahlquist barrier states that the trapezoidal rule is the most accurate amongst the A-stable linear multistep methods. More precisely, a linear multistep method that is A-stable has at most order two, and the error constant of a second-order A-stable linear multistep method cannot be better than the error constant of the trapezoidal rule. In fact, the region of absolute stability for the trapezoidal rule is precisely the left-half plane. This means that if the trapezoidal rule is applied to the linear test equation y''' = λy'', the numerical solution decays to zero if and only if the exact solution does. Notes

References 
 . . See also 
Crank–Nicolson method

Runge–Kutta methods