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.
Contents
Method
Suppose that we want to solve the differential equation
The trapezoidal rule is given by the formula
where
This is an implicit method: the value
Motivation
Integrating the differential equation from
The trapezoidal rule states that the integral on the right-hand side can be approximated as
Now combine both formulas and use that
Error analysis
It follows from the error analysis of the trapezoidal rule for quadrature that the local truncation error
Thus, the trapezoidal rule is a second-order method. This result can be used to show that the global error is
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.