home
PDF (letter size)
PDF (legal size)

Finite difference approximation formulas

Nasser M. Abbasi

July 2, 2015   Compiled on May 22, 2020 at 4:00am

Contents

1 Approximation to first derivative
2 Approximation to second derivative

1 Approximation to first derivative

These formulas below approximate \(u^{\prime }\) at \(x=x_{j}\) where \(j\) is the grid point number.






formula truncation Truncation common name and





error error order common notation





1 \(u_{j}^{\prime }\approx \frac{1}{h}\left ( u_{j+1}-u_{j}\right ) \) \(-u_{j}^{\prime \prime }\frac{h}{2}-u_{j}^{\left ( 3\right ) }\frac{h^{2}}{3!}-\cdots \) \(O\left ( h\right ) \) one point forward \(D_{+}\)





2 \(u_{j}^{\prime }\approx \frac{1}{h}\left ( u_{j}-u_{j-1}\right ) \) \(u_{j}^{\prime \prime }\frac{h}{2}-u_{j}^{\left ( 3\right ) }\frac{h^{2}}{3!}+\cdots \) \(O\left ( h\right ) \) one point backward \(D_{\_}\)





3 \(u_{j}^{\prime }\approx \frac{1}{2h}\left ( u_{j+1}-u_{j-1}\right ) \) \(-u_{j}^{\left ( 3\right ) }\frac{h^{2}}{6}-u_{j}^{\left ( 6\right ) }\frac{h^{5}}{6!}-\cdots \) \(O\left ( h^{2}\right ) \) centered difference, \(D_{0}=\frac{D_{+}+D_{\_}}{2}\)





4 \(u_{j}^{\prime }\approx \frac{1}{h}\left ( \frac{3}{2}u_{j}-2u_{j+1}+\frac{1}{2}u_{j+2}\right ) \) to do \(O\left ( h^{2}\right ) \) 3 points forward difference





5 \(u_{j}^{\prime }\approx \frac{1}{6}\left ( 2u_{j+1}+3u_{j}-6u_{j-1}+u_{j-2}\right ) \) to do \(O\left ( h^{3}\right ) \)





For example, to obtain the third formula above, we start from Taylor series and obtain\[ u_{j+1}=u_{j}+hu_{j}^{\prime }+\frac{h^{2}}{2!}u_{j}^{\prime \prime }+\frac{h^{3}}{3!}u_{j}^{\prime \prime \prime }+\cdots \] then we write it again for the previous point\[ u_{j-1}=u_{j}-hu_{j}^{\prime }+\frac{h^{2}}{2!}u_{j}^{\prime \prime }-\frac{h^{3}}{3!}u_{j}^{\prime \prime \prime }\cdots \] Notice the sign change in the expressions. We now subtract the second formula above from the above resulting in\[ u_{j+1}-u_{j-1}=2hu_{j}^{\prime }+2\frac{h^{3}}{3!}u_{j}^{\prime \prime \prime }+\cdots \] Or\begin{align*} u_{j+1}-u_{j-1} & =2hu_{j}^{\prime }+2\frac{h^{3}}{3!}u_{j}^{\prime \prime \prime }+\cdots \\ \frac{u_{j+1}-u_{j-1}}{2h} & =u_{j}^{\prime }+\overset{O(h^{2})\text{ error}}{\overbrace{h^{2}\frac{u_{j}^{\prime \prime \prime }}{3!}+\cdots }} \end{align*}

2 Approximation to second derivative

These formulas below approximate \(u^{\prime \prime }\) at \(x=x_{j}\) where \(j\) is the grid point number. For approximation to \(u^{\prime \prime }\) the accuracy of the approximation formula must be no less than \(2\).






formula truncation Truncation common name





error error order





1 \(u_{j}^{\prime \prime }\approx \frac{1}{h^{2}}\left ( U_{j-1}-2U_{j}+U_{j+1}\right ) \) \(-u^{\left ( 4\right ) }\frac{h^{2}}{12}-u^{\left ( 6\right ) }\frac{h^{4}}{360}-\cdots \) \(O\left ( h^{2}\right ) \) 3 points centered difference





To obtain the third formula above, we start from Taylor series. This results in\[ u_{j+1}=u_{j}+hu_{j}^{\prime }+\frac{h^{2}}{2!}u_{j}^{\prime \prime }+\frac{h^{3}}{3!}u_{j}^{\prime \prime \prime }+\frac{h^{4}}{4!}u_{j}^{\prime \prime \prime \prime }\cdots \] Then we write it again for the previous point\[ u_{j-1}=u_{j}-hu_{j}^{\prime }+\frac{h^{2}}{2!}u_{j}^{\prime \prime }-\frac{h^{3}}{3!}u_{j}^{\prime \prime \prime }+\frac{h^{4}}{4!}u_{j}^{\prime \prime \prime \prime }\cdots \] Notice the sign change in the expressions. We now add the second formula above from the above resulting in\begin{align*} u_{j+1}+u_{j-1} & =2u_{j}+2h^{2}u_{j}^{\prime \prime }+2\frac{h^{4}}{4!}u_{j}^{\prime \prime \prime \prime }+\cdots \\ \frac{u_{j-1}-2u_{j}+u_{j-1}}{2h^{2}} & =u_{j}^{\prime \prime }+\overset{O(h^{2})\text{ error}}{\overbrace{h^{2}\frac{u_{j}^{\prime \prime \prime }}{4!}+\cdots }} \end{align*}