contents.rst

Problem 1 - a

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 − 1ϕ_i − 1 + a_iϕ_i + a_i + 1ϕ_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 Γ 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: AΦ = Q. A can be described as a matrix or tensor form with two ranks. Φ and Q are vectors.