22. Introduction to Finite Volume Method#
\[\begin{split}
\begin{aligned}
\int
\dfrac{\mathrm{d}^{2}T}{\mathrm{d}x^{2}}
\mathrm{d}V&=
0.\\
\int_{w}^{e}
\dfrac{\mathrm{d}^{2}T}{\mathrm{d}x^{2}}
\mathrm{d}x&=
0.\\
{\left.\dfrac{\mathrm{d}T}{\mathrm{d}x}\right|}_{e}-
{\left.\dfrac{\mathrm{d}T}{\mathrm{d}x}\right|}_{w}&=
0.\\
\dfrac{T_{i+1}-T_{i}}{h}-
\dfrac{T_{i}-T_{i-1}}{h}&=
0.\\
\dfrac{T_{i+1}-2T_{i}+T_{i-1}}{h}&=
0.\\
\dfrac{T_{i+1}-T_{i}}{h}-
\dfrac{T_{i}-T_{i-1}}{h}&=
0.\\
T_{i}&=
\dfrac{1}{2}
\left(T_{i+1}+T_{i-1}\right).
\end{aligned}
\end{split}\]