home
PDF (letter size)
PDF (legal size)

## Finite diﬀerence approximation formulas

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

### 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*}