Convection of one species using different schemes

The unsteady convection equation for a concentration $c$ that is convected with velocity $v$ is:

For unsteady problems we will use the method of lines. After performing a spatial discretization the resulting set of ODE’s has the form:

where $\mathbf{c}$ is the column vector that contains the concentrations, $c_{i}$, where $i$ is the label of the grid cell.

You are asked to implement the one-component convection equation using the method-of-lines described above, using different spatial discretization methods, namely, upwind, central differencing and a total variance diminishing (TVD) scheme. For the time integration use a standard IVP solver.

Suggested approach:

• Use NumPy arrays to store the concentrations, $c_{i}$, and right hand side, $f_{i}$, at different axial positions $i$.

• Prefer using NumPy array indexing notation over for loops for element-wise operations. This approach leverages NumPy’s optimized performance and often results in more readable and efficient code.

• For each method, plot axial concentration profiles for several times.

• Compare your results with an analytical solution of the problem to verify your code.

• Play with the number of grid points $N$ used in the spatial discretization.

Questions:

1. Implement upwind discretization, where

2. Implement central differencing, where

3. Implement a TVD scheme, e.g. Min-Mod, for the convection. See lecture notes for the scheme.