2.1 Explicit Methods

2.1.1 FTCS

With FTCS, the forward time derivative, and the centered space derivative are used. Hence the advection PDE can be written as

\begin{equation} \frac {C_{i}^{n+1}-C_{i}^{n}}{\tau }=-u\left ( \frac {C_{i+1}^{n}-C_{i-1}^{n}}{2h}\right ) \tag {0}\end{equation}

Solving for \(C_{i}^{n+1}\) results in

\begin{equation} \fbox {$C_{i}^{n+1}=C_{i}^{n}-\frac {u\tau }{2h}\left ( C_{i+1}^{n}-C_{i-1}^{n}\right ) $} \tag {1}\end{equation}

This method will be shown to be unconditionally unstable.

2.1.2 Downwind

Here, the forward time derivative for\(\frac {\partial C}{\partial t}\) is used and also the forward space derivative for\(\frac {\partial C}{\partial x}\). This results in

\[ \frac {C_{i}^{n+1}-C_{i}^{n}}{\tau }=-u\left ( \frac {C_{i+1}^{n}-C_{i}^{n}}{h}\right ) \]

\[ \fbox {$C_{i}^{n+1}=C_{i}^{n}-\frac {u\tau }{h}\left ( C_{i+1}^{n}-C_{i}^{n}\right ) $}\]

This method will be shown to be unconditionally unstable as well.

2.1.3 Upwind

Here, the forward time derivative for \(\frac {\partial C}{\partial t}\) is used, and the backward derivative for \(\frac {\partial C}{\partial x}\) is used. This results in

\[ \frac {C_{i}^{n+1}-C_{i}^{n}}{\tau }=-u\left ( \frac {C_{i}^{n}-C_{i-1}^{n}}{h}\right ) \]

\[ \fbox {$C_{i}^{n+1}=C_{i}^{n}-\frac {u\tau }{h}\left ( C_{i}^{n}-C_{i-1}^{n}\right ) $}\]

This will be shown to be stable if \(\frac {u\tau }{h}\leq 1\)

2.1.4 LAX

Looking at the FTCS eq (1) above, and shown below again

\[ C_{i}^{n+1}=C_{i}^{n}-\frac {u\tau }{2h}\left ( C_{i+1}^{n}-C_{i-1}^{n}\right ) \]

The term \(C_{i}^{n}\) above is replaced by its average value\(\frac {C_{i+1}^{n}+C_{i-1}^{n}}{2}\) to obtain the LAX method

\begin{equation} \fbox {$C_{i}^{n+1}=\frac {1}{2}\left ( C_{i+1}^{n}+C_{i-1}^{n}\right ) -\frac {u\tau }{2h}\left ( C_{i+1}^{n}-C_{i-1}^{n}\right ) $} \tag {4}\end{equation}

This method will be shown to be stable if \(\frac {u\tau }{h}\leq 1\)

2.1.5 Lax-Wendroff

By using the second-order finite difference scheme for the time derivative, the method of Lax-Wendroff method is obtained

\[ C_{i}^{n+1}=C_{i}^{n}-\frac {u\tau }{2h}\left ( C_{i+1}^{n}-C_{i-1}^{n}\right ) +\frac {u^{2}\tau ^{2}}{2h^{2}}\left ( C_{i+1}^{n}+C_{i-1}^{n}-2C_{i}^{n}\right ) \]

2.1.6 Leap-frog

In this method, the centered derivative is used for both time and space. This results in

\[ \fbox {$\frac {C_{i}^{n+1}-C_{i}^{n-1}}{2\tau }=-u\left ( \frac {C_{i+1}^{n}-C_{i-1}^{n}}{2h}\right ) $}\]

This method requires a special starting procedure due to the term \(C_{i}^{n-1}\). Another scheme such as Lax can be used to kick start this method.