Problem 1 - aΒΆ

Develop a finite difference algorithm using central differences for the solution of the transport equation. Describe the essential steps.

Given equation:

\[U\frac{\partial \phi}{\partial x} = \frac{\partial}{\partial x} \left ( \Gamma \frac{\partial \phi}{\partial x} \right ) + Q\]

To make the given form of equation convenient to be converted into descritized algebraic equations, terms having dependent variables are sorted out in the left hand side leaving \(Q\) in the right hand side as:

\[U\frac{\partial \phi}{\partial x} - \frac{\partial}{\partial x} \left ( \Gamma \frac{\partial \phi}{\partial x} \right ) = + Q\]

Now, the left hand side is composed of convection term and diffusion term, respectively. These two term are going to be referred to as divergence term and laplacian term, respectively.

  • Divergence term descritization:

    \[U\frac{\partial \phi}{\partial x}:\;\; U\left ( \frac{\phi_{i+1} - \phi_{i-1}}{\Delta x} \right )\]

    Again, further descritzation should be done for the remaining first derivative in the middle points at \(i+1/2\) and \(i-1/2\) and it leads to:

    \[-\frac{1}{\Delta x} \left [ \frac{\Gamma_{i+1/2}\left ( \phi_{i+1} - \phi_{i-1} \right ) - \Gamma_{i-1/2}\left ( \phi_{i} - \phi_{i-1} \right )}{\Delta x} \right ]\]

  • Laplacian term descritization:

    \[- \frac{\partial}{\partial x} \left ( \Gamma \frac{\partial \phi}{\partial x} \right ): \;\; -\frac{1}{\Delta x} \left [ \left ( \Gamma \frac{\partial \phi}{\partial x} \right )_{i+1/2} - \left ( \Gamma \frac{\partial \phi}{\partial x} \right )_{i-1/2} \right ]\]

Constructing every terms with source term in right hand side at \(i\) node point becomes such that the final form has three terms with respective corresponding \(i\) node and neightbor points, \(i-1\) and \(i+1\) will become:

\[a_{i-1} \phi_{i-1} + a_{i} \phi_{i} + a_{i+1} \phi_{i+1} = Q_{i}\]

where,

\[a_{i} = \frac{\left ( \Gamma_{i+1/2} + \Gamma_{i-1/2} \right )}{\Delta x^2}\]
\[a_{i-1} = - \left ( \frac{\Gamma_{i-1/2}}{\Delta x^2} + \frac{U}{\Delta x} \right )\]
\[a_{i+1} = - \left ( \frac{\Gamma_{i+1/2}}{\Delta x^2} - \frac{U}{\Delta x} \right )\]

Here again we need to quantify the diffusion coefficient \(\Gamma\) at middle points where are not actually in presence. Therefore, linear interpolation is done for those middle point for having diffusivity in the second derivative terms:

\[\Gamma_{i+1/2} = \frac{1}{2}\left ( \Gamma_{i+1} + \Gamma_{i-1} \right )\]

Now we have single algebraic equation for each single node point. The node point is now linked to the neighbor right next to it. Thus, we can construct tri-diagonal matrix for those coefficients when we construct system of linear equations in 1-dimensional problem: \(A \Phi = Q\). \(A\) can be described as a matrix or tensor form with two ranks. \(\Phi\) and \(Q\) are vectors.