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Finite difference approximation formulas

Nasser M. Abbasi

July 2, 2015   Compiled on July 9, 2025 at 3:54am

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*}