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
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.
Integrating the differential equation from to , we find that
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:
The region of absolute stability for the trapezoidal rule is
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.
- Iserles, Arieh (1996), A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, ISBN 978-0-521-55655-2.
- Süli, Endre; Mayers, David (2003), An Introduction to Numerical Analysis, Cambridge University Press, ISBN 0521007941.