View source for Trapezoidal rule (differential equations)

{{short description|Numerical method for solving ordinary differential equations}}

In [[numerical analysis]] and [[scientific computing]], the '''trapezoidal rule''' is a [[numerical methods for ordinary differential equations|numerical method]] to solve [[ordinary differential equations]] derived from the [[trapezoidal rule]] for computing integrals. The trapezoidal rule is an [[implicit and explicit methods|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
<math display="block"> y' = f(t,y). </math>
The trapezoidal rule is given by the formula
<math display="block"> y_{n+1} = y_n + \tfrac 1 2 h \Big( f(t_n,y_n) + f(t_{n+1},y_{n+1}) \Big), </math>
where <math> h = t_{n+1} - t_n </math> is the step size.<ref>{{harvnb|Iserles|1996|p=8}}; {{harvnb|Süli|Mayers|2003|p=324}}</ref>

This is an implicit method: the value <math> y_{n+1} </math> 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.<ref>{{harvnb|Süli|Mayers|2003|p=324}}</ref> Cutting short, using only the guess from Eulers method is equivalent to performing [[Heun's method]].

== Motivation ==

Integrating the differential equation from <math> t_n </math> to <math> t_{n+1} </math>, we find that
<math display="block"> y(t_{n+1}) - y(t_n) = \int_{t_n}^{t_{n+1}} f(t,y(t)) \,\mathrm{d}t. </math>
The [[trapezoidal rule]] states that the integral on the right-hand side can be approximated as
<math display="block"> \int_{t_n}^{t_{n+1}} f(t,y(t)) \,\mathrm{d}t \approx \tfrac12 h \Big( f(t_n,y(t_n)) + f(t_{n+1},y(t_{n+1})) \Big). </math>
Now combine both formulas and use that <math> y_n \approx y(t_n) </math> and <math> y_{n+1} \approx y(t_{n+1}) </math> to get the trapezoidal rule for solving ordinary differential equations.<ref>{{harvnb|Iserles|1996|p=8}}; {{harvnb|Süli|Mayers|2003|p=324}}</ref>

== Error analysis ==

It follows from the error analysis of the trapezoidal rule for quadrature that the [[local truncation error]] <math> \tau_n </math> of the trapezoidal rule for solving differential equations can be bounded as:
<math display="block"> |\tau_n| \le \tfrac1{12} h^3 \max_t |y'''(t)|. </math>
Thus, the trapezoidal rule is a second-order method.{{Citation needed|date=September 2019}} This result can be used to show that the global error is <math> O(h^2) </math> as the step size <math> h </math> tends to zero (see [[big O notation]] for the meaning of this).<ref>{{harvnb|Iserles|1996|p=9}}; {{harvnb|Süli|Mayers|2003|p=325}}</ref>

== Stability ==

[[Image:Stability region for trapezoidal method.svg|thumb|The pink region is the stability region for the trapezoidal method.]]
The [[stiff equation|region of absolute stability]] for the trapezoidal rule is
<math display="block"> \{ z \in \mathbb{C} \mid \operatorname{Re}(z) < 0 \}. </math>
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.<ref>{{harvnb|Süli|Mayers|2003|p=324}}</ref>

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''' = λ''y'', the numerical solution decays to zero if and only if the exact solution does. However, the decay of the numerical solution can be many orders of magnitude slower than that of the true solution.

== Notes ==
{{reflist}}

== References ==
* {{Citation | last1=Iserles | first1=Arieh | author1-link=Arieh Iserles | title=A First Course in the Numerical Analysis of Differential Equations | publisher=[[Cambridge University Press]] | isbn=978-0-521-55655-2 | year=1996|url=https://books.google.com/books?id=7Zofw3SFTWIC&q=%22Trapezoidal+rule%22}}.
* {{Citation | last1=Süli | first1=Endre | last2=Mayers | first2=David | title=An Introduction to Numerical Analysis | publisher=[[Cambridge University Press]] | isbn=0521007941 | year=2003}}.

== See also ==
*[[Crank–Nicolson method]]


{{Numerical integrators}}

[[Category:Runge–Kutta methods]]