3.2. Truncation Error

The accuracy of the algebraic approximation to \(du/dt\) (3.3) can be determined with the help of the Taylor series expansion of \(u(t)\) around the point \(a=t^n\). Here, \(t^n\) is reprecenting a chosen time step. Note that \(u^{n+1}=u(t^{n+1})=u(t^n+\Delta t)\) and \(u^{n}=u(t^n)\).

(3.4)\[u(t)=u^n+\frac{u'(t^n)}{1!}(t-t^n)+\frac{u''(t^n)}{2!}(t-t^n)^2+\sum_{p=3}^{\infty}\frac{u^{(p)}(t^n)}{p!}(t-t^n)^p\]

Note that \(u'(t^n)\) can also be written \(\frac{du}{dt}|_{t^n}\). You will find that some references omit displaying the \(|_{t^n}\) in the derivative terms. We will include them here for completeness.

We are interested in the time step coming after \(t^n\). We, therefore, write out the equation for \(t=t^{n+1}\), and note that \((t^{n+1}-t^n)=\Delta t\):

(3.5)\[u^{n+1}=u^n+\Delta t\frac{du}{dt}|_{t^n}+\frac{\Delta t^2}{2!}\frac{d^2u}{dt^2}|_{t^n}+\sum_{p=3}^{\infty}\frac{\Delta t^p}{p!}\frac{d^{(p)}u}{dt^p}|_{t^n}\]

Rearranging for \(du/dt\), we obtain:

(3.6)\[\frac{du}{dt}|_{t^n}=\frac{u^{n+1}-u^n}{\Delta t}-\displaystyle\sum_{p=2}^{\infty} \frac{\Delta t^{p-1}}{p!}\frac{d^pu}{dt^p}|_{t^n}\]

The difference between the exact derivative (1.38) and our algebraic approximation (3.3) is the second term of the right hand side of (3.6). Expanding this term up to \(p=3\), we see its general form:

(3.7)\[\frac{\Delta t}{2!}\frac{d^2u}{dt^2}|_{t^n}+\frac{\Delta t^2}{3!}\frac{d^3u}{dt^3}|_{t^n}+O(\Delta t^3),\]

where \(O(\Delta t^3)\) represents terms proportional to \(\Delta t^3\) and to higher powers of \(\Delta t\).

Expression (3.7) is the truncation error, which is the error we incurr when we approximate the exact derivative (1.38) with the algebraic expression (3.3).

The algebraic approximation (3.3) can thus be written as:

(3.8)\[\frac{du}{dt}|_{t^n} = \frac{u^{n+1}-u^n}{\Delta t} + O(\Delta t),\]

where \(O(\Delta t)\) represents the truncation error of the approximation, whose first term, the leading order term is proportional to \(\Delta t\). Since the leading order term is proportional to the first power of \(\Delta t\), we say that this approximation is a first order approximation to the first derivative.