Transport Equations and the FTCS Scheme

Required Reading

Transport equation

Take a 1-dimensional PDE of the form

\begin{equation} \frac{\partial y}{\partial t} + v \frac{\partial y}{\partial x} = 0 \label{xport-equation} \end{equation}

for some constant $v$.

Integrate over some interval of $x$:

\begin{equation} \eqalign{ \int_{x_1}^{x_2}{\frac{\partial y}{\partial t}dx} &= \int_{x_1}^{x_2}{-v \frac{\partial y}{\partial x}dx} \cr \frac{\partial}{\partial t} \int_{x_1}^{x_2}{ydx} &= v (y(x_1, t) - y(x_2, t)). } \label{int-xport} \end{equation}

From this we can assert that equation \eqref{xport-equation} is flux-conservative; the change in $y$ inside $(x_1, x_2)$ at time $t$ is the amount flowing into the interval at $x_1$ minus the amount flowing out at $x_2$.

Let a solution to \eqref{xport-equation} be $y(x,t) = f(x-vt)$ so that

\begin{equation} y(x, 0) = y(x + vt, t), \label{xporting} \end{equation}

or in English, the value of $y$ at $x$ at some time $t$ is the same as the value of $y$ at $x + vt$, $t$ time units later. Or simply, the function is propagating in the positive $x$ direction with velocity $v$, hence the name Transport Equation.

Forward-time, Central-space

The Transport Equation can be solved numerically using forward-time and central-space finite differences.

First select $J > 0$ and $N > 0$. Discretise the problem; split the space $(x)$ dimension into $J+1$ points of size $\Delta x$, and the time $(t)$ dimension into $N+1$ steps of size $\Delta t$.

\begin{equation} \eqalign{ x_j &= x_0 + j\Delta x,\cr t_n &= t_0 + n\Delta t. } \label{discretise} \end{equation}

Notation: At this point, we introduce the special notation $y_j^n = y(t_n, x_j)$.

Recall the form of the forward finite difference; use it to estimate the time derivative:

\begin{equation} \frac{\partial y}{\partial t} = \frac{y_j^{n+1} - y_j^n}{\Delta t} + O(\Delta t). \label{forward-time} \end{equation}

Again, recall the form of the central finite difference; use it to estimate the space derivative:

\begin{equation} \frac{\partial y}{\partial x} = \frac{y_{j+1}^{n} - y_{j-1}^{n}}{2\Delta x} + O(\Delta x^2). \label{central-space} \end{equation}

Substitute equations \eqref{forward-time} and \eqref{central-space} back into the Transport Equation:

\begin{equation} \frac{y_j^{n+1} - y_j^n}{\Delta t} = -v \frac{y_{j+1}^{n} - y_{j-1}^{n}}{2\Delta x}. \label{xport-rewritten} \end{equation}

Solving equation \eqref{xport-rewritten} for $y_j^{n+1}$ yields the forward-time, central-space, or FTCS Scheme:

\begin{equation} y_j^{n+1} = y_j^n - \frac{v\Delta t}{2\Delta x} (y_{j+1}^{n} - y_{j-1}^{n}). \label{ftcs} \end{equation}

Conservation

Equation \eqref{ftcs} conserves $y(x,t)$. If boundary bounds are ignored, the following equality holds:

\begin{equation} \eqalign{ \sum_{j} y_j^{n+1} &= \sum_{j} y_j^n - \sum_{j} \frac{v\Delta t}{2\Delta x} (y_{j+1}^{n} - y_{j-1}^{n})\cr &= \sum_{j} y_j^n . } \label{conservation} \end{equation}

Explicit Form

If given initial values for $y_j^0, 0 \le j \le J$, the right hand side of equation \eqref{ftcs} is free of unknowns; the FTCS scheme is explicit.