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

$${\displaystyle y'=f(t,y).}$$

$${\displaystyle y_{n+1}=y_{n}+{\tfrac {1}{2))h{\Big (}f(t_{n},y_{n})+f(t_{n+1},y_{n+1}){\Big )},}$$

${\displaystyle h=t_{n+1}-t_{n))$

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.^[2] Cutting short, using only the guess from Eulers method is equivalent to performing Heun's method.

Motivation

Integrating the differential equation from ${\displaystyle t_{n))$ to ${\displaystyle t_{n+1))$, we find that

$${\displaystyle y(t_{n+1})-y(t_{n})=\int _{t_{n))^{t_{n+1))f(t,y(t))\,\mathrm {d} t.}$$

$${\displaystyle \int _{t_{n))^{t_{n+1))f(t,y(t))\,\mathrm {d} t\approx {\tfrac {1}{2))h{\Big (}f(t_{n},y(t_{n}))+f(t_{n+1},y(t_{n+1})){\Big )}.}$$

${\displaystyle y_{n}\approx y(t_{n})}$

${\displaystyle y_{n+1}\approx y(t_{n+1})}$

Error analysis

It follows from the error analysis of the trapezoidal rule for quadrature that the local truncation error ${\displaystyle \tau _{n))$ of the trapezoidal rule for solving differential equations can be bounded as:

$${\displaystyle |\tau _{n}|\leq {\tfrac {1}{12))h^{3}\max _{t}|y'''(t)|.}$$

${\displaystyle O(h^{2})}$

${\displaystyle h}$

Stability

The region of absolute stability for the trapezoidal rule is

$${\displaystyle \{z\in \mathbb {C} \mid \operatorname {Re} (z)<0\}.}$$

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.